Some inequalities for the q-beta and the q-gamma ... - Semantic Scholar
Applied Mathematics and Computation 204 (2008) 385–394
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Some inequalities for the q-beta and the q-gamma functions via some q-integral inequalities Ahmed Fitouhi a,*, Kamel Brahim b a b
Faculté des Sciences de Tunis, Campus Universitaire Tunis, El Menzah, 1060 Tunis, Tunisia Institut Préparatoire aux Études d’Ingénieur de Tunis, Tunisia
a r t i c l e
i n f o
a b s t r a c t Some new inequalities for the q-gamma, the q-beta and the q-analogue of the Psi functions are established via some q-integral inequalities. Ó 2008 Elsevier Inc. All rights reserved.
Keywords: q-Integral q-Gamma q-Beta
1. Introduction In classical analysis, integral inequalities have been well-developed and leading to a wide variety of applications in mathematics and physics (see [11–13] and references therein). In a survey paper [4], Dragomir et al. used certain clever integral inequalities to provide some interesting inequalities for the Euler’s beta and gamma functions. Interested by this type of inequalities, Agarwal et al. gave in [1] some improvements and generalizations of some of the Dragomir’s ones. In quantum-calculus, in spite of the natural difficulties, coming from the definition of the q-Jackson integral, the interest in the q-integral inequalities has grown in the last few years (see [2,6,14]). It is within this framework that this paper presents itself. The main object is provide some new q-integral inequalities and, as applications, we establish some inequalities for the q-beta and the q-gamma functions. This paper is organized as follows: in Section 2, we present some standard conventional notations and notions which will be used in the sequel. In Section 3, we state q-analogues of the Cebysev’s integral inequalities for synchronous (asynchronous) mappings and as a direct consequence, we give some inequalities involving the q-beta and the q-gamma functions. In Section 4, we establish some inequalities for these functions via q-Hölder’s integral inequality. Section 5 is devoted to some applications of the q-Grüss’ integral inequality. Finally, Section 6 shows a q-analogue of a Cebysev’s type inequality and gives some related applications for the q-beta and the q-gamma functions. 2. Notations and preliminaries For the convenience of the reader, we provide in this section a summary of the mathematical notations and definitions used in this paper. All of these results can be found in [5,8] or [9]. Throughout this paper, we will fix q 20; 1½. For a 2 C, we write
½aq ¼
1 qa ; 1q
ða; qÞn ¼
½nq ! ¼ ½1q ½2q . . . ½nq ;
n1 Y
ð1 aqk Þ;
n ¼ 1; 2; . . . ; 1;
k¼0
n 2 N:
* Corresponding author. E-mail addresses: [email protected] (A. Fitouhi), [email protected] (K. Brahim). 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.06.055
386
A. Fitouhi, K. Brahim / Applied Mathematics and Computation 204 (2008) 385–394
The q-derivative Dq f of a function f is given by
f ðxÞ f ðqxÞ ; ð1 qÞx
ðDq f ÞðxÞ ¼
if x 6¼ 0;
ð1Þ
and ðDq f Þð0Þ ¼ f 0 ð0Þ provided f 0 ð0Þ exists. The q-Jackson integrals from 0 to b and from 0 to 1 are defined by (see [7])
Z
b
f ðxÞdq x ¼ ð1 qÞb
0
1 X
f ðbq Þqn
n
ð2Þ
f ðqn Þqn ;
ð3Þ
n¼0
and
Z
1
f ðxÞdq x ¼ ð1 qÞ
0
1 X n¼1
provided the sums converge absolutely. The q-Jackson integral in a generic interval ½a; b is given by (see [7])
Z
b
f ðxÞdq x ¼
Z
a
b
f ðxÞdq x
0
Z
a
f ðxÞdq x:
ð4Þ
0
We denote by I one of the following sets:
Rq;þ ¼ fqn : n 2 Zg;
ð5Þ
n
½0; bq ¼ fbq : n 2 Ng;
b > 0;
k
½a; bq ¼ fbq : 0 6 k 6 ng;
ð6Þ
b > 0;
n
a ¼ bq ;
n 2 N;
ð7Þ
R
and we note I f ðxÞdq x the q-integral of f on the correspondent I. We recall the two q-analogues of the exponential function (see [5,15]) given by
Ezq ¼0 u0 ð; ; q; ð1 qÞzÞ ¼
1 X
q
nðn1Þ 2
n¼0
zn ¼ ðð1 qÞz; qÞ1 ½nq !
ð8Þ
and
ezq ¼1 u0 ð0; ; q; ð1 qÞzÞ ¼
1 X zn 1 : ¼ ðð1 qÞz; qÞ1 ½n ! q n¼0
ð9Þ
z z z z These q-exponential functions satisfy the following relations: Dq ezq ¼ ezq , Dq Ezq ¼ Eqz q and Eq eq ¼ eq Eq ¼ 1. The q-gamma function is defined by [7]
Cq ðxÞ ¼
ðq; qÞ1 ð1 qÞ1x ; ðqx ; qÞ1
x 6¼ 0; 1; 2; . . . ;
ð10Þ
it satisfies the following functional equation:
Cq ðx þ 1Þ ¼ ½xq Cq ðxÞ;
Cq ð1Þ ¼ 1;
ð11Þ
and having the following q-integral representation (see [9])
Cq ðxÞ ¼
Z
1 1q
0
t x1 Eqt q dq t;
x > 0:
ð12Þ
The previous q-integral representation, give that Cq is an infinitely differentiable function on 0; þ1½ and
CqðkÞ ðxÞ ¼
Z
1 1q
0
t x1 ðln tÞk Eqt q dq t;
x > 0;
k 2 N:
ð13Þ
The q-beta function is defined by (see [9])
Bq ðt; sÞ ¼
Z 0
1
xt1
ðxq; qÞ1 dq x; ðxqs ; qÞ1
s > 0;
t > 0:
ð14Þ
It satisfies
Bq ðt; sÞ ¼
Cq ðtÞCq ðsÞ ; Cq ðt þ sÞ
s > 0;
t > 0:
ð15Þ
A. Fitouhi, K. Brahim / Applied Mathematics and Computation 204 (2008) 385–394
387
3. q-Cebysev’s integral inequality and applications This section is devoted to state a q-analogue of the classical Cebysev’s integral inequality for synchronous (asynchronous) mappings and to give some related applications for the q-beta and the q-gamma functions. Definition 1. Let f and g be two functions defined on I. The functions f and g are said q-synchronous (q-asynchronous) on I if
ðf ðxÞ f ðyÞÞðgðxÞ gðyÞÞ P ð6Þ0 8x; y 2 I:
ð16Þ
Note that if f and g are both q-increasing or q-decreasing on I then they are q-synchronous on I. We begin by state the q-analogue of the Cebysev’s integral inequality. Proposition 1. Let f, g and h be three functions defined on I such that: (1) hðxÞ P 0; x 2 I, (2) f and g are q-synchronous (q-asynchronous) on I. Then
Z
hðxÞdq x
Z
I
hðxÞf ðxÞgðxÞdq x P ð6Þ
I
Z
hðxÞf ðxÞdq x
Z
I
hðxÞgðxÞdq x:
ð17Þ
I
Proof. We have
Z
hðxÞdq x
Z
I
hðxÞf ðxÞgðxÞdq x
Z
I
hðxÞf ðxÞdq x
I
Z
hðxÞgðxÞdq x ¼ 1=2
I
Z Z I
hðxÞhðyÞ½f ðxÞ f ðyÞ½gðxÞ gðyÞdq xdq y
I
So, the result follows from the conditions (1) and (2). h The following theorem is a direct consequence of the previous proposition. Theorem 1. Let m, n, p and p0 be some positive reals such that
ðp mÞðp0 nÞ 6 ðPÞ0: Then
Bq ðp; p0 ÞBq ðm; nÞ P ð6ÞBq ðp; nÞBq ðm; p0 Þ
ð18Þ
Cq ðp þ nÞCq ðp0 þ mÞ P ð6ÞCq ðp þ p0 ÞCq ðm þ nÞ:
ð19Þ
and
Proof. Fix m, n, p and p0 in 0; þ1½, satisfying the condition of the theorem and the functions f, g and h on ½0; 1q by
f ðxÞ ¼ xpm ;
gðxÞ ¼
ðxqn ; qÞ1 ðxqp0 ; qÞ1
ðxq; qÞ1 : ðxqn ; qÞ1
and hðxÞ ¼ xm1
ð20Þ
From the conditions
Dq f ðxÞ ¼ ½p mq xpm1
ð21Þ
and 0
Dq gðxÞ ¼ qp ½n p0 q
ðxqnþ1 ; qÞ1 ; ðxqp0 ; qÞ1
ð22Þ
one can see that f and g are q-synchronous (q-asynchronous) on ½0; 1q . So, by using Proposition 1, we obtain
Z
1
hðxÞdq x
0
Z
1
hðxÞf ðxÞgðxÞdq x P ð6Þ 0
Z
1
hðxÞf ðxÞdq x 0
Z
1
hðxÞgðxÞdq x:
ð23Þ
0
Thus,
Z 0
1
xm1
ðxq; qÞ1 dq x ðxqn ; qÞ1
Z 0
1
xp1
ðxq; qÞ1 dq x P ð6Þ ðxqp0 ; qÞ1
Z 0
1
xp1
ðxq; qÞ1 dq x ðxqn ; qÞ1
Z 0
1
xm1
ðxq; qÞ1 dq x; ðxqp0 ; qÞ1
ð24Þ
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A. Fitouhi, K. Brahim / Applied Mathematics and Computation 204 (2008) 385–394
which implies that
Bq ðm; nÞBq ðp; p0 Þ P ð6ÞBq ðp; nÞBq ðm; p0 Þ:
ð25Þ
Now, according to the relations (15) and (18), we obtain
Cq ðmÞCq ðnÞ Cq ðpÞCq ðp0 Þ Cq ðpÞCq ðnÞ Cq ðmÞCq ðp0 Þ P ð6Þ : 0 Cq ðm þ nÞ Cq ðp þ p Þ Cq ðp þ nÞ Cq ðm þ p0 Þ
ð26Þ
Therefore,
Cq ðp þ nÞCq ðp0 þ mÞ P ð6ÞCq ðp þ p0 ÞCq ðm þ nÞ:
ð27Þ
Corollary 1. For all p; m > 0, we have
Bq ðp; mÞ P ½Bq ðp; pÞBq ðm; mÞ1=2
ð28Þ
Cq ðp þ mÞ 6 ½Cq ð2pÞCq ð2mÞ1=2 :
ð29Þ
and
Proof. A direct application of Theorem 1, with p0 ¼ p and n ¼ m, gives the results.
h
Corollary 2. For all u; v > 0, we have
Cq
u þ v qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 Cq ðuÞCq ðvÞ: 2
ð30Þ
Proof. The inequality follows from (29), by taking p ¼ u2 and m ¼ 2v. h Theorem 2. Let m, p and k be real numbers satisfying m; p > 0 and p > k > m and let n be a nonnegative integer. If
kðp m kÞ P ð6Þ0
ð31Þ
ð2nÞ ð2nÞ Cqð2nÞ ðpÞCð2nÞ q ðmÞ P ð6ÞCq ðp kÞCq ðm þ kÞ:
ð32Þ
then
1 Proof. Let f, g and h be the functions defined on I ¼ ½0; 1q q by
f ðxÞ ¼ xpmk ;
2n and hðxÞ ¼ xm1 Eqx q ðln xÞ :
gðxÞ ¼ xk
We have
Dq f ðxÞ ¼ ½p m kq xpmk1
and Dq gðxÞ ¼ ½kq xk1 :
If the condition (31) holds, one can show that the functions f and g are q-synchronous (q-asynchronous) on I and Proposition 1 gives
Z
1 1q
0
2n xm1 Eqx q ðln xÞ dq x
ð6Þ
Z
1 1q
0
Z
1 1q
0
2n xpmk xk xm1 Eqx q ðln xÞ dq x P
2n xpmk xm1 Eqx q ðln xÞ dq x
Z 0
1 1q
2n xk xm1 Eqx q ðln xÞ dq x;
which is equivalent to
Z 0
1 1q
2n xm1 Eqx q ðln xÞ dq x
Z
1 1q
0
2n xp1 Eqx q ðln xÞ dq x P ð6Þ
Z 0
1 1q
2n xpk1 Eqx q ðln xÞ dq x
Z 0
1 1q
2n xkþm1 Eqx q ðln xÞ dq x:
Hence, the relation (13) completes the proof. h Taking n ¼ 0 in the previous theorem, we obtain the following result. Corollary 3. Let m, p and k be some real numbers under the conditions of Theorem 2, we have
Cq ðpÞCq ðmÞ P ð6ÞCq ðp kÞCq ðm þ kÞ
ð33Þ
A. Fitouhi, K. Brahim / Applied Mathematics and Computation 204 (2008) 385–394
389
and
Bq ðp; mÞ P ð6ÞBq ðp k; m þ kÞ:
ð34Þ
Corollary 4. Let n be a nonnegative integer, p > 0 and p0 2 R such that j p0 j< p. Then 0 ½Cqð2nÞ ðpÞ2 6 Cqð2nÞ ðp p0 ÞCð2nÞ q ðp þ p Þ:
ð35Þ
Proof. By choosing m ¼ p and k ¼ p0 , we obtain
kðp m kÞ ¼ ðp0 Þ2 6 0 and the result turns out from Theorem 2. h Taking in the previous result p ¼ uþv and p0 ¼ uv , we obtain the following result: 2 2 Corollary 5. Let u; v be two positive real numbers and n be a nonnegative integer. Then
Cð2nÞ q
u þ v 2
6
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2nÞ Cð2nÞ q ðuÞCq ðvÞ:
ð36Þ
Corollary 6. Let p > 0 and p0 2 R such that j p0 j< p. Then
C2q ðpÞ 6 Cq ðp p0 ÞCq ðp þ p0 Þ
ð37Þ
Bq ðp; pÞ 6 Bq ðp p0 ; p þ p0 Þ:
ð38Þ
and
Proof. For n ¼ 0, the inequality (35) becomes
C2q ðpÞ 6 Cq ðp p0 ÞCq ðp þ p0 Þ: The inequality (38) follows from (15). h Now, let us recall the definition (see [4]). Definition 2. The positive real numbers a and b may be called similarly (oppositely) unitary if
ða 1Þðb 1Þ P ð6Þ0: Now, we shall prove the following result: Theorem 3. If a; b > 0 be similarly (oppositely) unitary and n a nonnegative integer. Then ð2nÞ ð2nÞ ð2nÞ Cð2nÞ q ð2ÞCq ða þ bÞ P ð6ÞCq ða þ 1ÞCq ðb þ 1Þ:
ð39Þ
Proof. In Theorem 2, set m ¼ 2, p ¼ a þ b and k ¼ b 1. The condition (31) becomes
kðp m kÞ ¼ ða 1Þðb 1Þ P ð6Þ0:
ð40Þ
ð2nÞ ð2nÞ ð2nÞ Cð2nÞ q ð2ÞCq ða þ bÞ P ð6ÞCq ða þ 1ÞCq ðb þ 1Þ:
ð41Þ
So,
Corollary 7. If a; b > 0 and be similarly (oppositely) unitary. Then
Cq ða þ bÞ P ð6Þ½aq ½bq Cq ðaÞCq ðbÞ
ð42Þ
and
Bq ða; bÞ 6 ðPÞ
1 : ½aq ½bq
ð43Þ
Proof. The inequality (42) follows from the previous theorem by taking n ¼ 0 and using the facts that Cq ð2Þ ¼ 1, Cq ða þ 1Þ ¼ ½aq Cq ðaÞ and Cq ðb þ 1Þ ¼ ½bq Cq ðbÞ. Eq. (15) together with Eq. (42) gives Eq. (43). h
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A. Fitouhi, K. Brahim / Applied Mathematics and Computation 204 (2008) 385–394
Corollary 8. The function ln Cq is superadditive for x > 1, in the sense that
ln Cq ða þ bÞ P ln Cq ðaÞ þ ln Cq ðbÞ: Proof. For all a; b P 1, we have
ln Cq ða þ bÞ P ln½aq þ ln½bq þ ln Cq ðaÞ þ ln Cq ðbÞ P ln Cq ðaÞ þ ln Cq ðbÞ; which completes the proof.
h
Corollary 9. For a P 1 and n ¼ 1; 2; . . ., we have
Cq ðnaÞ P ½n 1qa !½aq2n1 ½Cq ðaÞn :
ð44Þ
Proof. We proceed by induction on n. It is clear that the inequality is true for n ¼ 1. Suppose that Eq. (44) holds for an integer n P 1 and let us prove it for n þ 1. By Eq. (42), we have
Cq ððn þ 1ÞaÞ ¼ Cq ðna þ aÞ P ½naq ½aq Cq ðnaÞCq ðaÞ
ð45Þ
and by hypothesis, we have
Cq ðnaÞ P ½n 1qa !½aq2n1 ½Cq ðaÞn :
ð46Þ
The use of the fact that ½naq ¼ ½nqa ½aq , gives nþ1 Cq ððn þ 1ÞaÞ P ½naq ½aq ½n 1qa !½a2n1 ½Cq ðaÞn Cq ðaÞ P ½nqa !½a2n : q q ½Cq ðaÞ
The inequality (44) is then true for n þ 1. h For a given real m > 0 and a nonnegative integer n, consider the mapping
Cq;m;n ðxÞ ¼
Cqð2nÞ ðx þ mÞ Cqð2nÞ ðmÞ
:
We have the following result. Corollary 10. The mapping Cq;m;n ð:Þ is supermultiplicative on ½0; 1Þ, in the sense
Cq;m;n ðx þ yÞ P Cq;m;n ðxÞCq;m;n ðyÞ: Proof. Fix x; y in ½0; 1Þ and put p ¼ x þ y þ m and k ¼ y. We have
yðx þ y þ m m yÞ ¼ xy P 0: So, the previous theorem leads to
Cqð2nÞ ðmÞCqð2nÞ ðx þ y þ mÞ P Cqð2nÞ ðx þ mÞCð2nÞ q ðy þ mÞ;
ð47Þ
which is equivalent to
Cq;m;n ðx þ yÞ P Cq;m;n ðxÞCq;m;n ðyÞ:
ð48Þ
This achieves the proof. h
4. Inequalities via the q-Hölder’s one We begin this section by recalling the q-analogue of the q-Hölder’s integral inequality proved in [2]. Lemma 1. Let p and p0 be two positive reals satisfying 1p þ p10 ¼ 1, and f and g be two functions defined on I. Then
Z Z 1p Z 10 p p p0 f ðxÞgðxÞdq x 6 jf ðxÞj d x jgðxÞj d x : q q I
I
I
Owing this lemma, one can establish some new inequalities involving the q-gamma and q-beta functions.
ð49Þ
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A. Fitouhi, K. Brahim / Applied Mathematics and Computation 204 (2008) 385–394
Theorem 4. Let n be a nonnegative integer, x; y be two positive real numbers and a; b be two nonnegative real numbers such that a þ b ¼ 1. Then a b ð2nÞ ð2nÞ Cð2nÞ q ðax þ byÞ 6 ½Cq ðxÞ ½Cq ðyÞ ;
that is, the mapping C
ð2nÞ q
ð50Þ
is logarithmically convex on ð0; 1Þ.
1 Proof. Consider the following functions defined on I ¼ ½0; 1q q ,
2n a f ðtÞ ¼ taðx1Þ ðEqt q ðln tÞ Þ
2n b and gðtÞ ¼ tbðy1Þ ðEqt q ðln tÞ Þ :
By application of the q-Hölder’s integral inequality, with p ¼ 1a, we get
Z
1 1q
0
2n taðx1Þ tbðy1Þ Eqt q ðln tÞ dq t
"Z
1 1q
6 0
#a 2n t aðx1Þ:ð1=aÞ Eqt q ðln tÞ dq t
"Z
1 1q
0
#b 2n tbðy1Þ:ð1=bÞ Eqt q ðln tÞ dq t
;
which is equivalent to
Z 0
1 1q
2n taxþby1 Eqt q ðln tÞ dq t 6
"Z
1 1q
0
2n t x1 Eqt q ðln tÞ dq t
#a "Z
1 1q
0
#b 2n t y1 Eqt q ðln tÞ dq t
:
Then, Eq. (50) is a direct consequence of Eq. (13). h Corollary 11. Let ðp; p0 Þ; ðm; m0 Þ 2 ð0; 1Þ2 such that p þ p0 ¼ m þ m0 and a; b P 0 with a þ b ¼ 1. Then, we have
Bq ðaðp; p0 Þ þ bðm; m0 ÞÞ 6 ½Bq ðp; p0 Þa ½Bq ðm; m0 Þb :
ð51Þ
Proof. On the one hand, we have 0
Bq ðaðp; p0 Þ þ bðm; m0 ÞÞ ¼ Bq ðap þ bm; ap0 þ bm Þ ¼
Cq ðap þ bmÞCq ðap0 þ bm0 Þ Cq ðap þ bmÞCq ðap0 þ bm0 Þ ¼ : Cq ðaðp þ p0 Þ þ bðm þ m0 ÞÞ Cq ðap þ bm þ ap0 þ bm0 Þ
Since p þ p0 ¼ m þ m0 and a þ b ¼ 1, we have
Cq ðaðp þ p0 Þ þ bðm þ m0 ÞÞ ¼ Cq ðp þ p0 Þ ¼ Cq ðm þ m0 Þ:
ð52Þ
On the other hand, from Theorem 4, with n ¼ 0, we obtain
Cq ðap þ bmÞ 6 ½Cq ðpÞa ½Cq ðmÞb ;
ð53Þ
and
a
Cq ðap0 þ bm0 Þ 6 Cq ðp0 Þ ½Cq ðm0 Þb :
ð54Þ
Thus
a
a
Cq ðap þ bmÞCq ðap0 þ bm0 Þ 6 Cq ðpÞCq ðp0 Þ
b
Cq ðmÞCq ðm0 Þ :
ð55Þ
From Eq. (52), we deduce that
Cq ðap þ bmÞCq ðap0 þ bm0 Þ Cq ðpÞCq ðp0 Þ 6 0 0 Cq ðaðp þ p Þ þ bðm þ m ÞÞ Cq ðp þ p0 Þ
Cq ðmÞCq ðm0 Þ b ; Cq ðm þ m0 Þ
which completes the proof. h Now, we recall that the logarithmic derivative of the q-gamma function is defined on ð0; 1Þ, by
Wq ðxÞ ¼
C0q ðxÞ : Cq ðxÞ
The following result gives some properties of the function Wq . Theorem 5. Wq is monotonic non-decreasing and concave on ð0; 1Þ. Proof. By taking n ¼ 0 in Theorem 4, we obtain
Cq ðax þ byÞ 6 ½Cq ðxÞa ½Cq ðyÞb ; for x; y > 0 and a; b P 0 such that a þ b ¼ 1.
ð56Þ
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A. Fitouhi, K. Brahim / Applied Mathematics and Computation 204 (2008) 385–394
So the function ln Cq is convex. Then the monotonicity of Wq follows from the relation
C0q ðxÞ d ½ln Cq ðxÞ ¼ ¼ Wq ðxÞ; dx Cq ðxÞ
x > 0:
On the other hand, since
ðq; qÞ1 ð1 qÞ1x ; ðqx ; qÞ1
Cq ðxÞ ¼
ð57Þ
we obtain, for x > 0, 1 1 1 X X X d qxþk xþk ½ln Cq ðxÞ ¼ lnð1 qÞ þ ln q ¼ lnð1 qÞ þ ln q q qðxþkÞn dx 1 qxþk n¼0 k¼0 k¼0 Z q x1 1 X qðnþ1Þx ln q t dq t: ¼ lnð1 qÞ þ ln q ¼ lnð1 qÞ þ nþ1 ð1 qÞ 1 q 1 t 0 n¼0
Wq ðxÞ ¼
Now, let x; y > 0 and a; b P 0 such that a þ b ¼ 1. Then
Wq ðax þ byÞ þ lnð1 qÞ ¼
ln q ð1 qÞ
Z 0
q
t axþby1 ln q dq t ¼ ð1 qÞ 1t
Z 0
q
t aðx1Þþbðy1Þ dq t: 1t
ð58Þ
Since the mapping x7!tx is convex on R for t 2 ð0; 1Þ, we have
taðx1Þþbðy1Þ 6 at x1 þ bt
y1
;
for t 2 ½0; qq ; x; y > 0:
Thus,
ln q ð1 qÞ
Z 0
q
Z q x1 Z q y1 t axþby1 ln q t ln q t dq t P a dq t þ b dq t : ð1 qÞ 0 1 t ð1 qÞ 0 1 t 1t
ð59Þ
According to the relations (58) and (59), we have
Wq ðax þ byÞ þ lnð1 qÞ P aðWq ðxÞ þ lnð1 qÞÞ þ bðWq ðyÞ þ lnð1 qÞÞ P aWq ðxÞ þ bWq ðyÞ þ lnð1 qÞ: This proves the concavity of the function Wq . h 5. Inequalities via the q-Grüss’s one In [6] Gauchman gave a q-analogue of the Grüss’ integral inequality namely Lemma 2. Assume that m 6 f ðxÞ 6 M, u 6 gðxÞ 6 U, for each x 2 ½a; b, where m; M; u; U are given real constants. Then
Z b Z b 1 1 Z b 1 f ðxÞgðxÞdq x f ðxÞdq x gðxÞdq x 6 ðM mÞðU uÞ: 2 4 b a a ðb aÞ a a
ð60Þ
As application of the previous inequality we state the following result Theorem 6. Let m; n > 0, we have
1 1 6 : Bq ðm þ 1; n þ 1Þ ½m þ 1q ½n þ 1q 4
ð61Þ
Remark that from the relations (15) and (11), the inequality (61) is equivalent to
jCq ðm þ n þ 2Þ Cq ðm þ 2ÞCq ðn þ 2Þj 6
1 ½m þ 1q ½n þ 1q Cq ðm þ n þ 2Þ: 4
Proof. Consider the functions
f ðxÞ ¼ xm ;
gðxÞ ¼
ðxq; qÞ1 ; ðxqnþ1 ; qÞ1
x 2 ½0; 1; m; n > 0:
We have
0 6 f ðxÞ 6 1 and 0 6 gðxÞ 6 1 8x 2 ½0; 1:
ð62Þ
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A. Fitouhi, K. Brahim / Applied Mathematics and Computation 204 (2008) 385–394
Then, using the q-Grüss’ integral inequality, we obtain
Z
1
xm
0
ðxq; qÞ1 dq x ðxqnþ1 ; qÞ1
Z
1
xm d q x
Z
0
1
0
1 ðxq; qÞ1 6 : d x q 4 nþ1 ðxq ; qÞ1
ð63Þ
The inequality (61) follows from the definition of the q-beta function and the following facts: R 1 ðxq;qÞ1 1 dq x ¼ Bq ð1; n þ 1Þ ¼ ½nþ1 . h 0 ðxqnþ1 ;qÞ
R1 0
1 xm dq x ¼ ½mþ1 and q
q
1
6. q- Cebysev’s type inequalities and q-beta and q-gamma functions We begin this section by recalling the following Cebysev’s type inequality:
Z Z b Z b Z b b hðxÞdx hðxÞf ðxÞgðxÞdx hðxÞf ðxÞdx hðxÞgðxÞdx a a a a 2 !2 3 Z b Z b Z b 6 kf k1 kgk1 4 hðxÞdx x2 hðxÞdx xhðxÞdx 5; a
a
ð64Þ
a
provided that h is positive and f, g are differentiable with bounded first derivatives on ða; bÞ. A q-analogue of this inequality is given in the following lemma. Lemma 3. Let f, g and h be three functions defined on I such that (1) hðxÞ > 0, for all x 2 I, (2) Dq ðf Þ and Dq ðgÞ are bounded on I. Then, provided the q-integrals converge, we have
Z Z Z Z hðxÞdq x hðxÞf ðxÞgðxÞdq x hðxÞf ðxÞdq x hðxÞgðxÞdq x I I I I "Z Z # Z 2
6 kDq f k1;I kDq gk1;I
x2 hðxÞdq x
hðxÞdq x
I
where kDq f k1;I ¼ supx2I jDq f ðxÞj
xhðxÞdq x
I
and
ð65Þ
;
I
kDq gk1;I ¼ supx2I jDq gðxÞj.
Proof. From the definitions of the q-Jackson’s integrals and the q-derivative, we have for all x; y 2 I such that y < x,
f ðxÞ f ðyÞ ¼
Z
x
Dq f ðtÞdq t
and gðxÞ gðyÞ ¼
y
Z
x
Dq gðtÞdq t:
y
So, for all x; y 2 I,
jf ðxÞ f ðyÞj 6 kDq f k1;I jx yj and jgðxÞ gðyÞj 6 kDq gk1;I jx yj:
ð66Þ
Then,
Z Z Z Z hðxÞdq x hðxÞf ðxÞgðxÞdq x hðxÞf ðxÞdq x hðxÞgðxÞdq x I I I I Z Z ¼ 1=2 hðxÞhðyÞ½f ðxÞ f ðyÞ½gðxÞ gðyÞdq xdq y ZI ZI 6 1=2 hðxÞhðyÞjf ðxÞ f ðyÞjjgðxÞ gðyÞjdq xdq y ðÞ I
I
6 1=2kDq f k1;I kDq gk1;I
Z a
b
Z
b
hðxÞhðyÞðx yÞ2 dq xdq y:
a
Finally, the identity
1=2
Z Z I
hðxÞhðyÞðx yÞ2 dq xdq y ¼
I
Z I
hðxÞdq x
Z I
x2 hðxÞdq x
Z
xhðxÞdq x
2 ð67Þ
I
completes the proof. h Remark. Taking account of the specificity of the interval I, the inequality ðÞ holds. As a direct application, one has the following theorem.
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A. Fitouhi, K. Brahim / Applied Mathematics and Computation 204 (2008) 385–394
Theorem 7. For m; n > 1 and r; s > 1, we have
jBq ðr þ 1; s þ 1ÞBq ðm þ r þ 1; n þ s þ 1Þ Bq ðm þ r þ 1; s þ 1ÞBq ðr þ 1; n þ s þ 1Þj 6 qsþ1 ½mq ½nq ½Bq ðr þ 3; s þ 1ÞBq ðr þ 1; s þ 1Þ B2q ðr þ 2; s þ 1Þ:
ð68Þ
Proof. Let f, g and h be the functions defined on I ¼ ½0; 1q by
f ðxÞ ¼ xm ;
gðxÞ ¼
ðxqsþ1 ; qÞ1 ; ðxqnþsþ1 ; qÞ1
hðxÞ ¼ xr
ðxq; qÞ1 : ðxqsþ1 ; qÞ1
It is easy to see that
Dq f ðxÞ ¼ ½mq xm1 ; kDq f k1;I 6 ½mq
and Dq gðxÞ ¼ qsþ1 ½nq
ðxqsþ2 ; qÞ1 ; ðxqnþsþ1 ; qÞ1
and kDq gk1;I 6 qsþ1 ½nq :
So, the previous lemma leads to
Z
Z 1 Z 1 Z 1 ðxq; qÞ1 ðxq; qÞ1 ðxq; qÞ1 rþm rþm ðxq; qÞ1 r d x x d x x d x x d x q q q q sþ1 nþsþ1 sþ1 nþsþ1 ðxq ; qÞ1 ðxq ; qÞ1 ðxq ; qÞ1 ðxq ; qÞ1 0 0 0 0 "Z # Z Z 2 1 1 1 ðxq; qÞ1 ðxq; qÞ1 ðxq; qÞ1 6 qsþ1 ½nq ½mq : xr dq x xrþ2 dq x xrþ1 dq x sþ1 sþ1 ðxq ; qÞ1 ðxq ; qÞ1 ðxqsþ1 ; qÞ1 0 0 0 1
xr
The result follows, then, from Eq. (14). h Taking r ¼ s ¼ 0 in Theorem 7, the following result holds. Corollary 12. For m; n > 1, we have
jBq ðm þ 1; n þ 1Þ
q½mq ½nq 1 1 j6 ½m þ 1q ½n þ 1q ½3q ½22q
ð69Þ
and
jCq ðm þ n þ 2Þ Cq ðm þ 2ÞCq ðn þ 2Þj 6
q½mq ½nq ½3q ½22q
½m þ 1q ½n þ 1q Cq ðm þ n þ 2Þ:
ð70Þ
References aric´, On some inequalities for beta and gamma functions via some classical inequalities, J. Inequal. Appl. 2005 5 (2005) [1] R.P. Agarwal, N. Elezovic´, J. Pec 593–613. [2] K. Brahim, N. Bettaibi, M. Sellami, Integral inequalities in quantum calculus, submitted for publication. [4] S.S. Dragomir, R.P. Agarwal, N.S. Barnett, Inequality for beta and gamma functions via some classical and new integral inequalities, J. Inequal. 5 (2) (2000) 103–165. [5] G. Gasper, M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and its Application, vol. 35, Cambridge University Press, Cambridge, UK, 1990. [6] H. Gauchman, Integral inequalities in q-calculus, Comput. Math. Appl. 47 (2004) 281–300. [7] F.H. Jackson, On a q-definite integrals, Quarterly J. Pure Appl. Math. 41 (1910) 193–203. [8] V.G. Kac, P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002. [9] T.H. Koornwinder, q-Special functions, a tutorial, in: M. Gerstenhaber, J. Stasheff (Eds.), Deformation Theory and Quantum Groups with Applications to Mathematical Physics, Contemporary Mathematics, vol. 134, American Mathematical Society, 1992. [11] D.S. Mitrinovic´, J.E. Pecˇaric´, A.M. Fink, Inequalities involving functions and their integrals and derivatives, Mathematics and Its Applications (East European Series), vol. 53, Kluwer Academic, Dordrecht, 1991. [12] D.S. Mitrinovic´, J.E. Pecˇaric´, A.M. Fink, Classical and New Inequalities in Analysis, Mathematics and Its Applications (East European Series), vol. 61, Kluwer Academic, Dordrecht, 1993. [13] J.E. Pecˇaric´, F. Proschan, Y.L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering, vol. 187, Academic Press, Massachusetts, 1992. , S.D. Marinkovic , M.S. Stankovic , The inequalities for some types of q-integrals, arXiv math: CA/0605208 v1 8 May 2006. [14] P.M. Rajkovic [15] A. De Sole, V.G. Kac, On Integral Representations of q-gamma and q-beta Functions, Department of Mathematics, MIT, Cambridge, USA.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Some inequalities for the q-beta and the q-gamma functions via some q-integral inequalities Ahmed Fitouhi a,*, Kamel Brahim b a b
Faculté des Sciences de Tunis, Campus Universitaire Tunis, El Menzah, 1060 Tunis, Tunisia Institut Préparatoire aux Études d’Ingénieur de Tunis, Tunisia
a r t i c l e
i n f o
a b s t r a c t Some new inequalities for the q-gamma, the q-beta and the q-analogue of the Psi functions are established via some q-integral inequalities. Ó 2008 Elsevier Inc. All rights reserved.
Keywords: q-Integral q-Gamma q-Beta
1. Introduction In classical analysis, integral inequalities have been well-developed and leading to a wide variety of applications in mathematics and physics (see [11–13] and references therein). In a survey paper [4], Dragomir et al. used certain clever integral inequalities to provide some interesting inequalities for the Euler’s beta and gamma functions. Interested by this type of inequalities, Agarwal et al. gave in [1] some improvements and generalizations of some of the Dragomir’s ones. In quantum-calculus, in spite of the natural difficulties, coming from the definition of the q-Jackson integral, the interest in the q-integral inequalities has grown in the last few years (see [2,6,14]). It is within this framework that this paper presents itself. The main object is provide some new q-integral inequalities and, as applications, we establish some inequalities for the q-beta and the q-gamma functions. This paper is organized as follows: in Section 2, we present some standard conventional notations and notions which will be used in the sequel. In Section 3, we state q-analogues of the Cebysev’s integral inequalities for synchronous (asynchronous) mappings and as a direct consequence, we give some inequalities involving the q-beta and the q-gamma functions. In Section 4, we establish some inequalities for these functions via q-Hölder’s integral inequality. Section 5 is devoted to some applications of the q-Grüss’ integral inequality. Finally, Section 6 shows a q-analogue of a Cebysev’s type inequality and gives some related applications for the q-beta and the q-gamma functions. 2. Notations and preliminaries For the convenience of the reader, we provide in this section a summary of the mathematical notations and definitions used in this paper. All of these results can be found in [5,8] or [9]. Throughout this paper, we will fix q 20; 1½. For a 2 C, we write
½aq ¼
1 qa ; 1q
ða; qÞn ¼
½nq ! ¼ ½1q ½2q . . . ½nq ;
n1 Y
ð1 aqk Þ;
n ¼ 1; 2; . . . ; 1;
k¼0
n 2 N:
* Corresponding author. E-mail addresses: [email protected] (A. Fitouhi), [email protected] (K. Brahim). 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.06.055
386
A. Fitouhi, K. Brahim / Applied Mathematics and Computation 204 (2008) 385–394
The q-derivative Dq f of a function f is given by
f ðxÞ f ðqxÞ ; ð1 qÞx
ðDq f ÞðxÞ ¼
if x 6¼ 0;
ð1Þ
and ðDq f Þð0Þ ¼ f 0 ð0Þ provided f 0 ð0Þ exists. The q-Jackson integrals from 0 to b and from 0 to 1 are defined by (see [7])
Z
b
f ðxÞdq x ¼ ð1 qÞb
0
1 X
f ðbq Þqn
n
ð2Þ
f ðqn Þqn ;
ð3Þ
n¼0
and
Z
1
f ðxÞdq x ¼ ð1 qÞ
0
1 X n¼1
provided the sums converge absolutely. The q-Jackson integral in a generic interval ½a; b is given by (see [7])
Z
b
f ðxÞdq x ¼
Z
a
b
f ðxÞdq x
0
Z
a
f ðxÞdq x:
ð4Þ
0
We denote by I one of the following sets:
Rq;þ ¼ fqn : n 2 Zg;
ð5Þ
n
½0; bq ¼ fbq : n 2 Ng;
b > 0;
k
½a; bq ¼ fbq : 0 6 k 6 ng;
ð6Þ
b > 0;
n
a ¼ bq ;
n 2 N;
ð7Þ
R
and we note I f ðxÞdq x the q-integral of f on the correspondent I. We recall the two q-analogues of the exponential function (see [5,15]) given by
Ezq ¼0 u0 ð; ; q; ð1 qÞzÞ ¼
1 X
q
nðn1Þ 2
n¼0
zn ¼ ðð1 qÞz; qÞ1 ½nq !
ð8Þ
and
ezq ¼1 u0 ð0; ; q; ð1 qÞzÞ ¼
1 X zn 1 : ¼ ðð1 qÞz; qÞ1 ½n ! q n¼0
ð9Þ
z z z z These q-exponential functions satisfy the following relations: Dq ezq ¼ ezq , Dq Ezq ¼ Eqz q and Eq eq ¼ eq Eq ¼ 1. The q-gamma function is defined by [7]
Cq ðxÞ ¼
ðq; qÞ1 ð1 qÞ1x ; ðqx ; qÞ1
x 6¼ 0; 1; 2; . . . ;
ð10Þ
it satisfies the following functional equation:
Cq ðx þ 1Þ ¼ ½xq Cq ðxÞ;
Cq ð1Þ ¼ 1;
ð11Þ
and having the following q-integral representation (see [9])
Cq ðxÞ ¼
Z
1 1q
0
t x1 Eqt q dq t;
x > 0:
ð12Þ
The previous q-integral representation, give that Cq is an infinitely differentiable function on 0; þ1½ and
CqðkÞ ðxÞ ¼
Z
1 1q
0
t x1 ðln tÞk Eqt q dq t;
x > 0;
k 2 N:
ð13Þ
The q-beta function is defined by (see [9])
Bq ðt; sÞ ¼
Z 0
1
xt1
ðxq; qÞ1 dq x; ðxqs ; qÞ1
s > 0;
t > 0:
ð14Þ
It satisfies
Bq ðt; sÞ ¼
Cq ðtÞCq ðsÞ ; Cq ðt þ sÞ
s > 0;
t > 0:
ð15Þ
A. Fitouhi, K. Brahim / Applied Mathematics and Computation 204 (2008) 385–394
387
3. q-Cebysev’s integral inequality and applications This section is devoted to state a q-analogue of the classical Cebysev’s integral inequality for synchronous (asynchronous) mappings and to give some related applications for the q-beta and the q-gamma functions. Definition 1. Let f and g be two functions defined on I. The functions f and g are said q-synchronous (q-asynchronous) on I if
ðf ðxÞ f ðyÞÞðgðxÞ gðyÞÞ P ð6Þ0 8x; y 2 I:
ð16Þ
Note that if f and g are both q-increasing or q-decreasing on I then they are q-synchronous on I. We begin by state the q-analogue of the Cebysev’s integral inequality. Proposition 1. Let f, g and h be three functions defined on I such that: (1) hðxÞ P 0; x 2 I, (2) f and g are q-synchronous (q-asynchronous) on I. Then
Z
hðxÞdq x
Z
I
hðxÞf ðxÞgðxÞdq x P ð6Þ
I
Z
hðxÞf ðxÞdq x
Z
I
hðxÞgðxÞdq x:
ð17Þ
I
Proof. We have
Z
hðxÞdq x
Z
I
hðxÞf ðxÞgðxÞdq x
Z
I
hðxÞf ðxÞdq x
I
Z
hðxÞgðxÞdq x ¼ 1=2
I
Z Z I
hðxÞhðyÞ½f ðxÞ f ðyÞ½gðxÞ gðyÞdq xdq y
I
So, the result follows from the conditions (1) and (2). h The following theorem is a direct consequence of the previous proposition. Theorem 1. Let m, n, p and p0 be some positive reals such that
ðp mÞðp0 nÞ 6 ðPÞ0: Then
Bq ðp; p0 ÞBq ðm; nÞ P ð6ÞBq ðp; nÞBq ðm; p0 Þ
ð18Þ
Cq ðp þ nÞCq ðp0 þ mÞ P ð6ÞCq ðp þ p0 ÞCq ðm þ nÞ:
ð19Þ
and
Proof. Fix m, n, p and p0 in 0; þ1½, satisfying the condition of the theorem and the functions f, g and h on ½0; 1q by
f ðxÞ ¼ xpm ;
gðxÞ ¼
ðxqn ; qÞ1 ðxqp0 ; qÞ1
ðxq; qÞ1 : ðxqn ; qÞ1
and hðxÞ ¼ xm1
ð20Þ
From the conditions
Dq f ðxÞ ¼ ½p mq xpm1
ð21Þ
and 0
Dq gðxÞ ¼ qp ½n p0 q
ðxqnþ1 ; qÞ1 ; ðxqp0 ; qÞ1
ð22Þ
one can see that f and g are q-synchronous (q-asynchronous) on ½0; 1q . So, by using Proposition 1, we obtain
Z
1
hðxÞdq x
0
Z
1
hðxÞf ðxÞgðxÞdq x P ð6Þ 0
Z
1
hðxÞf ðxÞdq x 0
Z
1
hðxÞgðxÞdq x:
ð23Þ
0
Thus,
Z 0
1
xm1
ðxq; qÞ1 dq x ðxqn ; qÞ1
Z 0
1
xp1
ðxq; qÞ1 dq x P ð6Þ ðxqp0 ; qÞ1
Z 0
1
xp1
ðxq; qÞ1 dq x ðxqn ; qÞ1
Z 0
1
xm1
ðxq; qÞ1 dq x; ðxqp0 ; qÞ1
ð24Þ
388
A. Fitouhi, K. Brahim / Applied Mathematics and Computation 204 (2008) 385–394
which implies that
Bq ðm; nÞBq ðp; p0 Þ P ð6ÞBq ðp; nÞBq ðm; p0 Þ:
ð25Þ
Now, according to the relations (15) and (18), we obtain
Cq ðmÞCq ðnÞ Cq ðpÞCq ðp0 Þ Cq ðpÞCq ðnÞ Cq ðmÞCq ðp0 Þ P ð6Þ : 0 Cq ðm þ nÞ Cq ðp þ p Þ Cq ðp þ nÞ Cq ðm þ p0 Þ
ð26Þ
Therefore,
Cq ðp þ nÞCq ðp0 þ mÞ P ð6ÞCq ðp þ p0 ÞCq ðm þ nÞ:
ð27Þ
Corollary 1. For all p; m > 0, we have
Bq ðp; mÞ P ½Bq ðp; pÞBq ðm; mÞ1=2
ð28Þ
Cq ðp þ mÞ 6 ½Cq ð2pÞCq ð2mÞ1=2 :
ð29Þ
and
Proof. A direct application of Theorem 1, with p0 ¼ p and n ¼ m, gives the results.
h
Corollary 2. For all u; v > 0, we have
Cq
u þ v qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 Cq ðuÞCq ðvÞ: 2
ð30Þ
Proof. The inequality follows from (29), by taking p ¼ u2 and m ¼ 2v. h Theorem 2. Let m, p and k be real numbers satisfying m; p > 0 and p > k > m and let n be a nonnegative integer. If
kðp m kÞ P ð6Þ0
ð31Þ
ð2nÞ ð2nÞ Cqð2nÞ ðpÞCð2nÞ q ðmÞ P ð6ÞCq ðp kÞCq ðm þ kÞ:
ð32Þ
then
1 Proof. Let f, g and h be the functions defined on I ¼ ½0; 1q q by
f ðxÞ ¼ xpmk ;
2n and hðxÞ ¼ xm1 Eqx q ðln xÞ :
gðxÞ ¼ xk
We have
Dq f ðxÞ ¼ ½p m kq xpmk1
and Dq gðxÞ ¼ ½kq xk1 :
If the condition (31) holds, one can show that the functions f and g are q-synchronous (q-asynchronous) on I and Proposition 1 gives
Z
1 1q
0
2n xm1 Eqx q ðln xÞ dq x
ð6Þ
Z
1 1q
0
Z
1 1q
0
2n xpmk xk xm1 Eqx q ðln xÞ dq x P
2n xpmk xm1 Eqx q ðln xÞ dq x
Z 0
1 1q
2n xk xm1 Eqx q ðln xÞ dq x;
which is equivalent to
Z 0
1 1q
2n xm1 Eqx q ðln xÞ dq x
Z
1 1q
0
2n xp1 Eqx q ðln xÞ dq x P ð6Þ
Z 0
1 1q
2n xpk1 Eqx q ðln xÞ dq x
Z 0
1 1q
2n xkþm1 Eqx q ðln xÞ dq x:
Hence, the relation (13) completes the proof. h Taking n ¼ 0 in the previous theorem, we obtain the following result. Corollary 3. Let m, p and k be some real numbers under the conditions of Theorem 2, we have
Cq ðpÞCq ðmÞ P ð6ÞCq ðp kÞCq ðm þ kÞ
ð33Þ
A. Fitouhi, K. Brahim / Applied Mathematics and Computation 204 (2008) 385–394
389
and
Bq ðp; mÞ P ð6ÞBq ðp k; m þ kÞ:
ð34Þ
Corollary 4. Let n be a nonnegative integer, p > 0 and p0 2 R such that j p0 j< p. Then 0 ½Cqð2nÞ ðpÞ2 6 Cqð2nÞ ðp p0 ÞCð2nÞ q ðp þ p Þ:
ð35Þ
Proof. By choosing m ¼ p and k ¼ p0 , we obtain
kðp m kÞ ¼ ðp0 Þ2 6 0 and the result turns out from Theorem 2. h Taking in the previous result p ¼ uþv and p0 ¼ uv , we obtain the following result: 2 2 Corollary 5. Let u; v be two positive real numbers and n be a nonnegative integer. Then
Cð2nÞ q
u þ v 2
6
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2nÞ Cð2nÞ q ðuÞCq ðvÞ:
ð36Þ
Corollary 6. Let p > 0 and p0 2 R such that j p0 j< p. Then
C2q ðpÞ 6 Cq ðp p0 ÞCq ðp þ p0 Þ
ð37Þ
Bq ðp; pÞ 6 Bq ðp p0 ; p þ p0 Þ:
ð38Þ
and
Proof. For n ¼ 0, the inequality (35) becomes
C2q ðpÞ 6 Cq ðp p0 ÞCq ðp þ p0 Þ: The inequality (38) follows from (15). h Now, let us recall the definition (see [4]). Definition 2. The positive real numbers a and b may be called similarly (oppositely) unitary if
ða 1Þðb 1Þ P ð6Þ0: Now, we shall prove the following result: Theorem 3. If a; b > 0 be similarly (oppositely) unitary and n a nonnegative integer. Then ð2nÞ ð2nÞ ð2nÞ Cð2nÞ q ð2ÞCq ða þ bÞ P ð6ÞCq ða þ 1ÞCq ðb þ 1Þ:
ð39Þ
Proof. In Theorem 2, set m ¼ 2, p ¼ a þ b and k ¼ b 1. The condition (31) becomes
kðp m kÞ ¼ ða 1Þðb 1Þ P ð6Þ0:
ð40Þ
ð2nÞ ð2nÞ ð2nÞ Cð2nÞ q ð2ÞCq ða þ bÞ P ð6ÞCq ða þ 1ÞCq ðb þ 1Þ:
ð41Þ
So,
Corollary 7. If a; b > 0 and be similarly (oppositely) unitary. Then
Cq ða þ bÞ P ð6Þ½aq ½bq Cq ðaÞCq ðbÞ
ð42Þ
and
Bq ða; bÞ 6 ðPÞ
1 : ½aq ½bq
ð43Þ
Proof. The inequality (42) follows from the previous theorem by taking n ¼ 0 and using the facts that Cq ð2Þ ¼ 1, Cq ða þ 1Þ ¼ ½aq Cq ðaÞ and Cq ðb þ 1Þ ¼ ½bq Cq ðbÞ. Eq. (15) together with Eq. (42) gives Eq. (43). h
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Corollary 8. The function ln Cq is superadditive for x > 1, in the sense that
ln Cq ða þ bÞ P ln Cq ðaÞ þ ln Cq ðbÞ: Proof. For all a; b P 1, we have
ln Cq ða þ bÞ P ln½aq þ ln½bq þ ln Cq ðaÞ þ ln Cq ðbÞ P ln Cq ðaÞ þ ln Cq ðbÞ; which completes the proof.
h
Corollary 9. For a P 1 and n ¼ 1; 2; . . ., we have
Cq ðnaÞ P ½n 1qa !½aq2n1 ½Cq ðaÞn :
ð44Þ
Proof. We proceed by induction on n. It is clear that the inequality is true for n ¼ 1. Suppose that Eq. (44) holds for an integer n P 1 and let us prove it for n þ 1. By Eq. (42), we have
Cq ððn þ 1ÞaÞ ¼ Cq ðna þ aÞ P ½naq ½aq Cq ðnaÞCq ðaÞ
ð45Þ
and by hypothesis, we have
Cq ðnaÞ P ½n 1qa !½aq2n1 ½Cq ðaÞn :
ð46Þ
The use of the fact that ½naq ¼ ½nqa ½aq , gives nþ1 Cq ððn þ 1ÞaÞ P ½naq ½aq ½n 1qa !½a2n1 ½Cq ðaÞn Cq ðaÞ P ½nqa !½a2n : q q ½Cq ðaÞ
The inequality (44) is then true for n þ 1. h For a given real m > 0 and a nonnegative integer n, consider the mapping
Cq;m;n ðxÞ ¼
Cqð2nÞ ðx þ mÞ Cqð2nÞ ðmÞ
:
We have the following result. Corollary 10. The mapping Cq;m;n ð:Þ is supermultiplicative on ½0; 1Þ, in the sense
Cq;m;n ðx þ yÞ P Cq;m;n ðxÞCq;m;n ðyÞ: Proof. Fix x; y in ½0; 1Þ and put p ¼ x þ y þ m and k ¼ y. We have
yðx þ y þ m m yÞ ¼ xy P 0: So, the previous theorem leads to
Cqð2nÞ ðmÞCqð2nÞ ðx þ y þ mÞ P Cqð2nÞ ðx þ mÞCð2nÞ q ðy þ mÞ;
ð47Þ
which is equivalent to
Cq;m;n ðx þ yÞ P Cq;m;n ðxÞCq;m;n ðyÞ:
ð48Þ
This achieves the proof. h
4. Inequalities via the q-Hölder’s one We begin this section by recalling the q-analogue of the q-Hölder’s integral inequality proved in [2]. Lemma 1. Let p and p0 be two positive reals satisfying 1p þ p10 ¼ 1, and f and g be two functions defined on I. Then
Z Z 1p Z 10 p p p0 f ðxÞgðxÞdq x 6 jf ðxÞj d x jgðxÞj d x : q q I
I
I
Owing this lemma, one can establish some new inequalities involving the q-gamma and q-beta functions.
ð49Þ
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Theorem 4. Let n be a nonnegative integer, x; y be two positive real numbers and a; b be two nonnegative real numbers such that a þ b ¼ 1. Then a b ð2nÞ ð2nÞ Cð2nÞ q ðax þ byÞ 6 ½Cq ðxÞ ½Cq ðyÞ ;
that is, the mapping C
ð2nÞ q
ð50Þ
is logarithmically convex on ð0; 1Þ.
1 Proof. Consider the following functions defined on I ¼ ½0; 1q q ,
2n a f ðtÞ ¼ taðx1Þ ðEqt q ðln tÞ Þ
2n b and gðtÞ ¼ tbðy1Þ ðEqt q ðln tÞ Þ :
By application of the q-Hölder’s integral inequality, with p ¼ 1a, we get
Z
1 1q
0
2n taðx1Þ tbðy1Þ Eqt q ðln tÞ dq t
"Z
1 1q
6 0
#a 2n t aðx1Þ:ð1=aÞ Eqt q ðln tÞ dq t
"Z
1 1q
0
#b 2n tbðy1Þ:ð1=bÞ Eqt q ðln tÞ dq t
;
which is equivalent to
Z 0
1 1q
2n taxþby1 Eqt q ðln tÞ dq t 6
"Z
1 1q
0
2n t x1 Eqt q ðln tÞ dq t
#a "Z
1 1q
0
#b 2n t y1 Eqt q ðln tÞ dq t
:
Then, Eq. (50) is a direct consequence of Eq. (13). h Corollary 11. Let ðp; p0 Þ; ðm; m0 Þ 2 ð0; 1Þ2 such that p þ p0 ¼ m þ m0 and a; b P 0 with a þ b ¼ 1. Then, we have
Bq ðaðp; p0 Þ þ bðm; m0 ÞÞ 6 ½Bq ðp; p0 Þa ½Bq ðm; m0 Þb :
ð51Þ
Proof. On the one hand, we have 0
Bq ðaðp; p0 Þ þ bðm; m0 ÞÞ ¼ Bq ðap þ bm; ap0 þ bm Þ ¼
Cq ðap þ bmÞCq ðap0 þ bm0 Þ Cq ðap þ bmÞCq ðap0 þ bm0 Þ ¼ : Cq ðaðp þ p0 Þ þ bðm þ m0 ÞÞ Cq ðap þ bm þ ap0 þ bm0 Þ
Since p þ p0 ¼ m þ m0 and a þ b ¼ 1, we have
Cq ðaðp þ p0 Þ þ bðm þ m0 ÞÞ ¼ Cq ðp þ p0 Þ ¼ Cq ðm þ m0 Þ:
ð52Þ
On the other hand, from Theorem 4, with n ¼ 0, we obtain
Cq ðap þ bmÞ 6 ½Cq ðpÞa ½Cq ðmÞb ;
ð53Þ
and
a
Cq ðap0 þ bm0 Þ 6 Cq ðp0 Þ ½Cq ðm0 Þb :
ð54Þ
Thus
a
a
Cq ðap þ bmÞCq ðap0 þ bm0 Þ 6 Cq ðpÞCq ðp0 Þ
b
Cq ðmÞCq ðm0 Þ :
ð55Þ
From Eq. (52), we deduce that
Cq ðap þ bmÞCq ðap0 þ bm0 Þ Cq ðpÞCq ðp0 Þ 6 0 0 Cq ðaðp þ p Þ þ bðm þ m ÞÞ Cq ðp þ p0 Þ
Cq ðmÞCq ðm0 Þ b ; Cq ðm þ m0 Þ
which completes the proof. h Now, we recall that the logarithmic derivative of the q-gamma function is defined on ð0; 1Þ, by
Wq ðxÞ ¼
C0q ðxÞ : Cq ðxÞ
The following result gives some properties of the function Wq . Theorem 5. Wq is monotonic non-decreasing and concave on ð0; 1Þ. Proof. By taking n ¼ 0 in Theorem 4, we obtain
Cq ðax þ byÞ 6 ½Cq ðxÞa ½Cq ðyÞb ; for x; y > 0 and a; b P 0 such that a þ b ¼ 1.
ð56Þ
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So the function ln Cq is convex. Then the monotonicity of Wq follows from the relation
C0q ðxÞ d ½ln Cq ðxÞ ¼ ¼ Wq ðxÞ; dx Cq ðxÞ
x > 0:
On the other hand, since
ðq; qÞ1 ð1 qÞ1x ; ðqx ; qÞ1
Cq ðxÞ ¼
ð57Þ
we obtain, for x > 0, 1 1 1 X X X d qxþk xþk ½ln Cq ðxÞ ¼ lnð1 qÞ þ ln q ¼ lnð1 qÞ þ ln q q qðxþkÞn dx 1 qxþk n¼0 k¼0 k¼0 Z q x1 1 X qðnþ1Þx ln q t dq t: ¼ lnð1 qÞ þ ln q ¼ lnð1 qÞ þ nþ1 ð1 qÞ 1 q 1 t 0 n¼0
Wq ðxÞ ¼
Now, let x; y > 0 and a; b P 0 such that a þ b ¼ 1. Then
Wq ðax þ byÞ þ lnð1 qÞ ¼
ln q ð1 qÞ
Z 0
q
t axþby1 ln q dq t ¼ ð1 qÞ 1t
Z 0
q
t aðx1Þþbðy1Þ dq t: 1t
ð58Þ
Since the mapping x7!tx is convex on R for t 2 ð0; 1Þ, we have
taðx1Þþbðy1Þ 6 at x1 þ bt
y1
;
for t 2 ½0; qq ; x; y > 0:
Thus,
ln q ð1 qÞ
Z 0
q
Z q x1 Z q y1 t axþby1 ln q t ln q t dq t P a dq t þ b dq t : ð1 qÞ 0 1 t ð1 qÞ 0 1 t 1t
ð59Þ
According to the relations (58) and (59), we have
Wq ðax þ byÞ þ lnð1 qÞ P aðWq ðxÞ þ lnð1 qÞÞ þ bðWq ðyÞ þ lnð1 qÞÞ P aWq ðxÞ þ bWq ðyÞ þ lnð1 qÞ: This proves the concavity of the function Wq . h 5. Inequalities via the q-Grüss’s one In [6] Gauchman gave a q-analogue of the Grüss’ integral inequality namely Lemma 2. Assume that m 6 f ðxÞ 6 M, u 6 gðxÞ 6 U, for each x 2 ½a; b, where m; M; u; U are given real constants. Then
Z b Z b 1 1 Z b 1 f ðxÞgðxÞdq x f ðxÞdq x gðxÞdq x 6 ðM mÞðU uÞ: 2 4 b a a ðb aÞ a a
ð60Þ
As application of the previous inequality we state the following result Theorem 6. Let m; n > 0, we have
1 1 6 : Bq ðm þ 1; n þ 1Þ ½m þ 1q ½n þ 1q 4
ð61Þ
Remark that from the relations (15) and (11), the inequality (61) is equivalent to
jCq ðm þ n þ 2Þ Cq ðm þ 2ÞCq ðn þ 2Þj 6
1 ½m þ 1q ½n þ 1q Cq ðm þ n þ 2Þ: 4
Proof. Consider the functions
f ðxÞ ¼ xm ;
gðxÞ ¼
ðxq; qÞ1 ; ðxqnþ1 ; qÞ1
x 2 ½0; 1; m; n > 0:
We have
0 6 f ðxÞ 6 1 and 0 6 gðxÞ 6 1 8x 2 ½0; 1:
ð62Þ
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Then, using the q-Grüss’ integral inequality, we obtain
Z
1
xm
0
ðxq; qÞ1 dq x ðxqnþ1 ; qÞ1
Z
1
xm d q x
Z
0
1
0
1 ðxq; qÞ1 6 : d x q 4 nþ1 ðxq ; qÞ1
ð63Þ
The inequality (61) follows from the definition of the q-beta function and the following facts: R 1 ðxq;qÞ1 1 dq x ¼ Bq ð1; n þ 1Þ ¼ ½nþ1 . h 0 ðxqnþ1 ;qÞ
R1 0
1 xm dq x ¼ ½mþ1 and q
q
1
6. q- Cebysev’s type inequalities and q-beta and q-gamma functions We begin this section by recalling the following Cebysev’s type inequality:
Z Z b Z b Z b b hðxÞdx hðxÞf ðxÞgðxÞdx hðxÞf ðxÞdx hðxÞgðxÞdx a a a a 2 !2 3 Z b Z b Z b 6 kf k1 kgk1 4 hðxÞdx x2 hðxÞdx xhðxÞdx 5; a
a
ð64Þ
a
provided that h is positive and f, g are differentiable with bounded first derivatives on ða; bÞ. A q-analogue of this inequality is given in the following lemma. Lemma 3. Let f, g and h be three functions defined on I such that (1) hðxÞ > 0, for all x 2 I, (2) Dq ðf Þ and Dq ðgÞ are bounded on I. Then, provided the q-integrals converge, we have
Z Z Z Z hðxÞdq x hðxÞf ðxÞgðxÞdq x hðxÞf ðxÞdq x hðxÞgðxÞdq x I I I I "Z Z # Z 2
6 kDq f k1;I kDq gk1;I
x2 hðxÞdq x
hðxÞdq x
I
where kDq f k1;I ¼ supx2I jDq f ðxÞj
xhðxÞdq x
I
and
ð65Þ
;
I
kDq gk1;I ¼ supx2I jDq gðxÞj.
Proof. From the definitions of the q-Jackson’s integrals and the q-derivative, we have for all x; y 2 I such that y < x,
f ðxÞ f ðyÞ ¼
Z
x
Dq f ðtÞdq t
and gðxÞ gðyÞ ¼
y
Z
x
Dq gðtÞdq t:
y
So, for all x; y 2 I,
jf ðxÞ f ðyÞj 6 kDq f k1;I jx yj and jgðxÞ gðyÞj 6 kDq gk1;I jx yj:
ð66Þ
Then,
Z Z Z Z hðxÞdq x hðxÞf ðxÞgðxÞdq x hðxÞf ðxÞdq x hðxÞgðxÞdq x I I I I Z Z ¼ 1=2 hðxÞhðyÞ½f ðxÞ f ðyÞ½gðxÞ gðyÞdq xdq y ZI ZI 6 1=2 hðxÞhðyÞjf ðxÞ f ðyÞjjgðxÞ gðyÞjdq xdq y ðÞ I
I
6 1=2kDq f k1;I kDq gk1;I
Z a
b
Z
b
hðxÞhðyÞðx yÞ2 dq xdq y:
a
Finally, the identity
1=2
Z Z I
hðxÞhðyÞðx yÞ2 dq xdq y ¼
I
Z I
hðxÞdq x
Z I
x2 hðxÞdq x
Z
xhðxÞdq x
2 ð67Þ
I
completes the proof. h Remark. Taking account of the specificity of the interval I, the inequality ðÞ holds. As a direct application, one has the following theorem.
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Theorem 7. For m; n > 1 and r; s > 1, we have
jBq ðr þ 1; s þ 1ÞBq ðm þ r þ 1; n þ s þ 1Þ Bq ðm þ r þ 1; s þ 1ÞBq ðr þ 1; n þ s þ 1Þj 6 qsþ1 ½mq ½nq ½Bq ðr þ 3; s þ 1ÞBq ðr þ 1; s þ 1Þ B2q ðr þ 2; s þ 1Þ:
ð68Þ
Proof. Let f, g and h be the functions defined on I ¼ ½0; 1q by
f ðxÞ ¼ xm ;
gðxÞ ¼
ðxqsþ1 ; qÞ1 ; ðxqnþsþ1 ; qÞ1
hðxÞ ¼ xr
ðxq; qÞ1 : ðxqsþ1 ; qÞ1
It is easy to see that
Dq f ðxÞ ¼ ½mq xm1 ; kDq f k1;I 6 ½mq
and Dq gðxÞ ¼ qsþ1 ½nq
ðxqsþ2 ; qÞ1 ; ðxqnþsþ1 ; qÞ1
and kDq gk1;I 6 qsþ1 ½nq :
So, the previous lemma leads to
Z
Z 1 Z 1 Z 1 ðxq; qÞ1 ðxq; qÞ1 ðxq; qÞ1 rþm rþm ðxq; qÞ1 r d x x d x x d x x d x q q q q sþ1 nþsþ1 sþ1 nþsþ1 ðxq ; qÞ1 ðxq ; qÞ1 ðxq ; qÞ1 ðxq ; qÞ1 0 0 0 0 "Z # Z Z 2 1 1 1 ðxq; qÞ1 ðxq; qÞ1 ðxq; qÞ1 6 qsþ1 ½nq ½mq : xr dq x xrþ2 dq x xrþ1 dq x sþ1 sþ1 ðxq ; qÞ1 ðxq ; qÞ1 ðxqsþ1 ; qÞ1 0 0 0 1
xr
The result follows, then, from Eq. (14). h Taking r ¼ s ¼ 0 in Theorem 7, the following result holds. Corollary 12. For m; n > 1, we have
jBq ðm þ 1; n þ 1Þ
q½mq ½nq 1 1 j6 ½m þ 1q ½n þ 1q ½3q ½22q
ð69Þ
and
jCq ðm þ n þ 2Þ Cq ðm þ 2ÞCq ðn þ 2Þj 6
q½mq ½nq ½3q ½22q
½m þ 1q ½n þ 1q Cq ðm þ n þ 2Þ:
ð70Þ
References aric´, On some inequalities for beta and gamma functions via some classical inequalities, J. Inequal. Appl. 2005 5 (2005) [1] R.P. Agarwal, N. Elezovic´, J. Pec 593–613. [2] K. Brahim, N. Bettaibi, M. Sellami, Integral inequalities in quantum calculus, submitted for publication. [4] S.S. Dragomir, R.P. Agarwal, N.S. Barnett, Inequality for beta and gamma functions via some classical and new integral inequalities, J. Inequal. 5 (2) (2000) 103–165. [5] G. Gasper, M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and its Application, vol. 35, Cambridge University Press, Cambridge, UK, 1990. [6] H. Gauchman, Integral inequalities in q-calculus, Comput. Math. Appl. 47 (2004) 281–300. [7] F.H. Jackson, On a q-definite integrals, Quarterly J. Pure Appl. Math. 41 (1910) 193–203. [8] V.G. Kac, P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002. [9] T.H. Koornwinder, q-Special functions, a tutorial, in: M. Gerstenhaber, J. Stasheff (Eds.), Deformation Theory and Quantum Groups with Applications to Mathematical Physics, Contemporary Mathematics, vol. 134, American Mathematical Society, 1992. [11] D.S. Mitrinovic´, J.E. Pecˇaric´, A.M. Fink, Inequalities involving functions and their integrals and derivatives, Mathematics and Its Applications (East European Series), vol. 53, Kluwer Academic, Dordrecht, 1991. [12] D.S. Mitrinovic´, J.E. Pecˇaric´, A.M. Fink, Classical and New Inequalities in Analysis, Mathematics and Its Applications (East European Series), vol. 61, Kluwer Academic, Dordrecht, 1993. [13] J.E. Pecˇaric´, F. Proschan, Y.L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering, vol. 187, Academic Press, Massachusetts, 1992. , S.D. Marinkovic , M.S. Stankovic , The inequalities for some types of q-integrals, arXiv math: CA/0605208 v1 8 May 2006. [14] P.M. Rajkovic [15] A. De Sole, V.G. Kac, On Integral Representations of q-gamma and q-beta Functions, Department of Mathematics, MIT, Cambridge, USA.
