Coefficient bounds for p-valent functions - Semantic Scholar
Applied Mathematics and Computation 187 (2007) 35–46 www.elsevier.com/locate/amc
Coefficient bounds for p-valent functions Rosihan M. Ali *, V. Ravichandran, N. Seenivasagan School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia
Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday
Abstract Sharp bounds for japþ2 la2pþ1 j and jap+3j are derived for certain p-valent analytic functions. These are applied to obtain Fekete-Szego¨ like inequalities for several classes of functions defined by convolution. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Analytic functions; Starlike functions; Convex functions; p-Valent functions; Subordination; Convolution; Fekete-Szego¨ inequalities
1. Introduction Let Ap be the class of all functions of the form 1 X an z n f ðzÞ ¼ zp þ
ð1Þ
n¼pþ1
which are analytic P1in the open unit disk D :¼ {z : jzj < 1} and let A :¼ A1 . For f(z) given by (1) and g(z) given by gðzÞ ¼ zp þ n¼pþ1 bn zn , their convolution (or Hadamard product), denoted by f * g, is defined by 1 X an bn z n : ðf gÞðzÞ :¼ zp þ n¼pþ1
The function f(z) is subordinate to the function g(z), written f(z) g(z), provided there is an analytic function w(z) defined on D with w(0) = 0 and jw(z)j < 1 such that f(z) = g(w(z)). Let u be an analytic function with positive real part in the unit disk D with u(0) = 1 and u 0 (0) > 0 that maps D onto a region starlike with respect to 1 and symmetric with respect to the real axis. We define the class S b;p ðuÞ to be the subclass of Ap consisting of functions f(z) satisfying
*
Corresponding author. E-mail addresses: [email protected] (R.M. Ali), [email protected] (V. Ravichandran), [email protected] (N. Seenivasagan).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.08.100
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1 1 zf 0 ðzÞ 1 uðzÞ; 1þ b p f ðzÞ As special cases, let S p ðuÞ :¼ S 1;p ðuÞ;
ðz 2 D and b 2 C n f0gÞ:
S b ðuÞ :¼ S b;1 ðuÞ;
S ðuÞ :¼ S 1;1 ðuÞ:
For a fixed analytic function g 2 Ap with positive coefficients, define the class S b;p;g ðuÞ to be the class of all functions f 2 Ap satisfying f g 2 S b;p ðuÞ. ThisPclass includes as special cases several other classes studied 1 in the literature. For example, when gðzÞ ¼ zp þ n¼pþ1 np zn , the class S b;p;g ðuÞ reduces to the class Cb,p(u) consisting of functions f 2 Ap satisfying 1 1 zf 00 ðzÞ 1þ 0 1 þ uðzÞ; ðz 2 D and b 2 C n f0gÞ: b bp f ðzÞ The classes S*(u) and C(u) :¼ C1,1(u) were introduced and studied by Ma and Minda [9]. Define the class Rb,p(u) to be the class of all functions f 2 Ap satisfying 1 f 0 ðzÞ 1þ 1 uðzÞ; ðz 2 D and b 2 C n f0gÞ; b pzp1 and for a fixed function g with positive coefficients, let Rb,p,g(u) be the class of all functions f 2 Ap satisfying f * g 2 Rb,p(u). Several authors [4,8,10,16,17,12] have studied the classes of analytic functions defined by using the expres0 2 00 ðzÞ sion zff ðzÞ þ a z ffðzÞðzÞ. We shall also consider a class defined by the corresponding quantity for p-valent functions. Define the class S p ða; uÞ to be the class of all functions f 2 Ap satisfying 1 þ að1 pÞ zf 0 ðzÞ a z2 f 00 ðzÞ þ uðzÞ p f ðzÞ p f ðzÞ
ðz 2 D and a P 0Þ:
Note that S p ð0; uÞ is the class S p ðuÞ. Let S p;g ða; uÞ be the class of all functions f 2 Ap for which f g 2 S p ða; uÞ. We shall also consider the class LM p ða; uÞ consisting of p-valent a-convex functions with respect to u. These are functions f 2 Ap satisfying 1 a zf 0 ðzÞ a zf 00 ðzÞ þ 1þ 0 uðzÞ ðz 2 D and a P 0Þ: p f ðzÞ p f ðzÞ Further let Mp(a, u) be the class of functions f 2 Ap satisfying a 1a 1 zf 0 ðzÞ zf 00 ðzÞ 1þ 0 uðzÞ ðz 2 D and a P 0Þ: p f ðzÞ f ðzÞ Functions in this class are called logarithmic p-valent a-convex functions with respect to u. In this paper, we obtain Fekete-Szego¨ inequalities and bounds for the coefficient ap+3 for the classes S p ðuÞ and S p;g ðuÞ. These results are then extended to the other classes defined earlier. See [1–7,9,13,14,18] for FeketeSzego¨ problem for certain related classes of functions. Let X be the class of analytic functions of the form wðzÞ ¼ w1 z þ w2 z2 þ
ð2Þ
in the unit disk D satisfying the condition jw(z)j < 1. We need the following lemmas to prove our main results. Lemma 1. If w 2 X, then 8 > < t if t 6 1; 2 jw2 tw1 j 6 1 if 1 6 t 6 1; > : t if t P 1: When t < 1 or t > 1, equality holds if and only if w(z) = z or one of its rotations. If 1 < t < 1, then equality holds if and only if w(z) = z2 or one of its rotations. Equality holds for t = 1 if and only if wðzÞ ¼
R.M. Ali et al. / Applied Mathematics and Computation 187 (2007) 35–46
37
kþz kþz z 1þkz ð0 6 k 6 1Þ or one of its rotations while for t = 1, equality holds if and only if wðzÞ ¼ z 1þkz ð0 6 k 6 1Þ or one of its rotations. Also the sharp upper bound above can be improved as follows when 1 < t < 1:
jw2 tw21 j þ ðt þ 1Þjw1 j2 6 1 ð1 < t 6 0Þ and jw2 tw21 j þ ð1 tÞjw1 j2 6 1 ð0 < t < 1Þ: Lemma 1 is a reformulation of a Lemma of Ma and Minda [9]. Lemma 2 [5, Inequality 7, p. 10]. If w 2 X, then, for any complex number t, jw2 tw21 j 6 maxf1; jtjg: The result is sharp for the functions w(z) = z2 or w(z) = z. Lemma 3 [11]. If w 2 X, then for any real numbers q1 and q2, the following sharp estimate holds: jw3 þ q1 w1 w2 þ q2 w31 j 6 H ðq1 ; q2 Þ;
ð3Þ
where 8 1 > > > > > > > > > jq2 j > > > > > > > 12 < jq1 jþ1 2 H ðq1 ; q2 Þ ¼ 3 ðjq1 j þ 1Þ 3ðjq1 jþ1þq2 Þ > > > > 2 2 12 > > q1 4 q1 4 1 > > q > 3 2 q21 4q2 3ðq2 1Þ > > > > > 12 > > jq1 j1 : 2 ðjq j 1Þ 3
1
3ðjq1 j1q2 Þ
for ðq1 ; q2 Þ 2 D1 [ D2 ; for ðq1 ; q2 Þ 2
7 S
Dk ;
k¼3
for ðq1 ; q2 Þ 2 D8 [ D9 ; for ðq1 ; q2 Þ 2 D10 [ D11 f2; 1g; for ðq1 ; q2 Þ 2 D12 :
The extremal functions, up to rotations, are of the form wðzÞ ¼ z3 ;
wðzÞ ¼ z;
wðzÞ ¼ w1 ðzÞ ¼ je1 j ¼ je2 j ¼ 1;
zðt1 zÞ ; 1 t1 z
wðzÞ ¼ w0 ðzÞ ¼
zð½ð1 kÞe2 þ ke1 e1 e2 zÞ ; 1 ½ð1 kÞe1 þ ke2 z
wðzÞ ¼ w2 ðzÞ ¼ ih0
zðt2 þ zÞ ; 1 þ t2 z ih0
e1 ¼ t0 e 2 ða bÞ; e2 ¼ e 2 ðia bÞ; rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h0 h0 ba ; b ¼ 1 t20 sin2 ; k¼ a ¼ t0 cos ; 2b 2 2 1 12 2q2 ðq21 þ 2Þ 3q21 2 jq1 j þ 1 t0 ¼ ; t1 ¼ ; 3ðjq1 j þ 1 þ q2 Þ 3ðq2 1Þðq21 4q2 Þ 12 jq1 j 1 h0 q1 q2 ðq21 þ 8Þ 2ðq21 þ 2Þ t2 ¼ ; cos ¼ : 3ðjq1 j 1 q2 Þ 2 2 2q2 ðq21 þ 2Þ 3q21
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The sets Dk, k = 1, 2, . . . , 12, are defined as follows:
1 D1 ¼ ðq1 ; q2 Þ : jq1 j 6 ; jq2 j 6 1 ; 2
1 4 D2 ¼ ðq1 ; q2 Þ : 6 jq1 j 6 2; ðjq1 j þ 1Þ3 ðjq1 j þ 1Þ 6 q2 6 1 ; 2 27
1 D3 ¼ ðq1 ; q2 Þ : jq1 j 6 ; q2 6 1 ; 2
1 2 D4 ¼ ðq1 ; q2 Þ : jq1 j P ; q2 6 ðjq1 j þ 1Þ ; 2 3 D5 ¼ fðq1 ; q2 Þ : jq1 j 6 2; q2 P 1g;
1 D6 ¼ ðq1 ; q2 Þ : 2 6 jq1 j 6 4; q2 P ðq21 þ 8Þ ; 12
2 D7 ¼ ðq1 ; q2 Þ : jq1 j P 4; q2 P ðjq1 j 1Þ ; 3
1 2 4 D8 ¼ ðq1 ; q2 Þ : 6 jq1 j 6 2; ðjq1 j þ 1Þ 6 q2 6 ðjq1 j þ 1Þ3 ðjq1 j þ 1Þ ; 2 3 27
2 2jq jðjq j þ 1Þ D9 ¼ ðq1 ; q2 Þ : jq1 j P 2; ðjq1 j þ 1Þ 6 q2 6 2 1 1 ; 3 q1 þ 2jq1 j þ 4
2jq1 jðjq1 j þ 1Þ 1 2 6 q2 6 ðq1 þ 8Þ ; D10 ¼ ðq1 ; q2 Þ : 2 6 jq1 j 6 4; 2 q1 þ 2jq1 j þ 4 12
2jq jðjq j þ 1Þ 2jq jðjq j 1Þ 6 q2 6 2 1 1 D11 ¼ ðq1 ; q2 Þ : jq1 j P 4; 2 1 1 ; q1 þ 2jq1 j þ 4 q1 2jq1 j þ 4
2jq jðjq j 1Þ 2 6 q2 6 ðjq1 j 1Þ : D12 ¼ ðq1 ; q2 Þ : jq1 j P 4; 2 1 1 q1 2jq1 j þ 4 3 2. Coefficient bounds By making use of the Lemmas 1–3, we prove the following bounds for the class S p;g ðuÞ: Theorem 1. Let u(z) = 1 + B1z + B2z2 + B3z3 + , and r1 :¼
B2 B1 þ pB21 ; 2pB21
r2 :¼
B2 þ B1 þ pB21 ; 2pB21
If f(z) given by (1) belongs to S p ðuÞ, then
8p 2 > < 2 B2 þ ð1 2lÞpB1 apþ2 la2pþ1 6 pB2 1 >
: p 2 B2 þ ð1 2lÞpB21
r3 :¼
B2 þ pB21 : 2pB21
if l 6 r1 ; if r1 6 l 6 r2 ; if l P r2 :
Further, if r1 6 l 6 r3, then 1 B2 pB 2 2 1 þ ð2l 1ÞpB1 japþ1 j 6 1 : apþ2 lapþ1 þ 2pB1 B1 2 If r3 6 l 6 r2, then 1 B2 pB 2 2 1 þ ð2l 1ÞpB1 japþ1 j 6 1 : apþ2 lapþ1 þ 2pB1 B1 2
ð4Þ
ð5Þ
ð6Þ
R.M. Ali et al. / Applied Mathematics and Computation 187 (2007) 35–46
For any complex number l,
pB B2 apþ2 la2pþ1 6 1 max 1; þ ð1 2lÞpB1 : 2 B1 Further, pB japþ3 j 6 1 H ðq1 ; q2 Þ; 3 where H(q1, q2) is as defined in Lemma 3, q1 :¼
4B2 þ 3pB21 2B1
and
q2 :¼
39
ð7Þ
ð8Þ
2B3 þ 3pB1 B2 þ p2 B31 : 2B1
These results are sharp. Proof. If f ðzÞ 2 S p ðuÞ; then there is an analytic function w(z) = w1z + w2z2 + 2 X such that zf 0 ðzÞ ¼ uðwðzÞÞ: pf ðzÞ Since zf 0 ðzÞ apþ1 ¼1þ zþ pf ðzÞ p
ð9Þ ! a2pþ1 2apþ2 2 þ z þ p p
! a3pþ1 3 3apþ3 3 apþ1 apþ2 þ z þ ; p p p
we have from (9), apþ1 ¼ pB1 w1 ; 1 pB1 w2 þ pðB2 þ pB21 Þw21 apþ2 ¼ 2 and apþ3
ð10Þ ð11Þ
pB1 4B2 þ 3pB21 2B3 þ 3pB1 B2 þ p2 B31 3 w3 þ ¼ w1 w2 þ w1 : 3 2B1 2B1
ð12Þ
Using (10) and (11), we have pB ð13Þ apþ2 la2pþ1 ¼ 1 w2 vw21 ; 2 where B2 v :¼ pB1 ð2l 1Þ : B1 The results (4)–(6) are established by an application of Lemma 1, inequality (7) by Lemma 2 and (8) follows from Lemma 3. To show that the bounds in (4)–(6) are sharp, we define the functions Kun (n = 2, 3, . . .) by zK 0un ðzÞ ¼ uðzn1 Þ; pK un ðzÞ
0
K un ð0Þ ¼ 0 ¼ ½K un ð0Þ 1
and the functions Fk and Gk (0 6 k 6 1) by zF 0k ðzÞ zðz þ kÞ ¼u ; F k ð0Þ ¼ 0 ¼ F 0k ð0Þ 1 pF k ðzÞ 1 þ kz and zG0k ðzÞ zðz þ kÞ ¼u ; Gk ð0Þ ¼ 0 ¼ G0k ð0Þ 1: pGk ðzÞ 1 þ kz Clearly the functions K un ; F k ; Gk 2 S p ðuÞ. We shall also write Ku :¼ Ku2.
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R.M. Ali et al. / Applied Mathematics and Computation 187 (2007) 35–46
If l < r1 or l > r2, then equality holds if and only if f is Ku or one of its rotations. When r1 < l < r2, then equality holds if and only if f is Ku3 or one of its rotations. If l = r1 then equality holds if and only if f is Fk or one of its rotations. Equality holds for l = r2 if and only if f is Gk or one of its rotations. h Corollary 1. Let u(z) = 1 + B1z + B2z2 + B3z3 + , and let r1 :¼
g2pþ1 B2 B1 þ pB21 ; gpþ2 2pB21
r2 :¼
g2pþ1 B2 þ B1 þ pB21 ; gpþ2 2pB21
If f(z) given by (1) belongs to S p;g ðuÞ, then 8 gpþ2 > p 2 > > > 2gpþ2 B2 þ 1 2 g2pþ1 l B1 > > < pB1 apþ2 la2pþ1 6 2gpþ2 > > > > gpþ2 > p > B þ 1 2 l B21 : 2gpþ2 2 g2
r3 :¼
g2pþ1 B2 þ pB21 : gpþ2 2pB21
if l 6 r1 ; if r1 6 l 6 r2 ; if l P r2 :
pþ1
Further, if r1 6 l 6 r3, then apþ2 la2pþ1 þ
g2pþ1 2gpþ2 pB1
B2 1 þ B1
! ! gpþ2 pB1 2 2 2 l 1 B1 japþ1 j 6 : gpþ1 2gpþ2
If r3 6 l 6 r2, then apþ2 la2pþ1 þ
g2pþ1 2gpþ2 pB1
B2 1þ B1
! ! gpþ2 pB1 2 2 2 l 1 B1 japþ1 j 6 : gpþ1 2gpþ2
For any complex number l, ( B pB 2 1 max 1; þ apþ2 la2pþ1 6 B1 2gpþ2
! ) gpþ2 1 2 2 l B1 : gpþ1
Further, japþ3 j 6
pB1 H ðq1 ; q2 Þ; 3gpþ3
ð14Þ
where H(q1, q2) is as defined in Lemma 3, q1 :¼
4B2 þ 3pB21 2B1
and
q2 :¼
2B3 þ 3pB1 B2 þ p2 B31 : 2B1
These results are sharp. Theorem 2. Let u be as in Theorem 1. If f(z) given by (1) belongs to S b;p;g ðuÞ, then, for any complex number l, we have ( ! ) B pjbjB g2pþ1 2 1 2 max 1; þ bp 1 2 l B1 : apþ2 lapþ1 6 B1 2gpþ2 gpþ2 The result is sharp. Proof. The proof is similar to the proof of Theorem 1. h Proceeding similarly, we now obtain coefficient bounds for the class Rb,p,g(u).
R.M. Ali et al. / Applied Mathematics and Computation 187 (2007) 35–46
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Theorem 3. Let u(z) = 1 + B1z + B2z2 + B3z3 + If f(z) given by (1) belongs to Rb,p(u), then for any complex number l, ð15Þ apþ2 la2pþ1 6 jcj max f1; jvjg; where v :¼ l
pbB1 ðp þ 2Þ ðp þ 1Þ
2
B2 B1
and
c :¼
bpB1 : 2þp
Further, japþ3 j 6
jbjpB1 H ðq1 ; q2 Þ; 3þp
ð16Þ
where H(q1, q2) is as defined in Lemma 3, q1 :¼
2B2 B1
and
q2 :¼
B3 : B1
These results are sharp. Proof. A computation shows that 1 f 0 ðzÞ pþ1 pþ2 pþ3 apþ1 z þ apþ2 z2 þ apþ3 z3 þ : 1þ 1 ¼1þ b pzp1 bp bp bp Thus bpB1 w2 vw21 ¼ c w2 vw21 ; apþ2 la2pþ1 ¼ pþ2 h i 1 lð2þpÞ 1 where v :¼ bpBðpþ1Þ BB21 and c :¼ bpB . The result now follows from Lemmas 2 and 3. 2 pþ2
ð17Þ h
Remark 1. When p = 1 and 1 þ Az ð1 6 B 6 A 6 1Þ; uðzÞ ¼ 1 þ Bz inequality (15) reduces to give the inequality [3, Theorem 4, p. 894]. For the class Rb,p,g(u), we have the following result. Theorem 4. Let u(z) = 1 + B1z + B2z2 + B3z3 + . If f(z) given by (1) belongs to Rb,p(u), then for any complex number l, apþ2 la2pþ1 6 jcj max f1; jvjg; where v :¼
lg2pþ1 pbB1 ðp þ 2Þ B2 gpþ2 ðp þ 1Þ2 B1
and
c :¼
bpB1 : gpþ2 ð2 þ pÞ
Further, japþ3 j 6
jbjpB1 H ðq1 ; q2 Þ; ð3 þ pÞgpþ3
where H(q1, q2) is as defined in Lemma 3, q1 :¼
2B2 B1
and
These results are sharp.
q2 :¼
B3 : B1
ð18Þ
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We now prove the coefficient bounds for S p;g ða; uÞ. Theorem 5. Let u(z) = 1 + B1z + B2z2 + B3z3 + Further let r1 :¼
ð1 þ aðp þ 1ÞÞ½ð1 þ aðp þ 1ÞÞðB2 B1 Þ þ pB21 ; 2pB21
r2 :¼
ð1 þ aðp þ 1ÞÞ½ð1 þ aðp þ 1ÞÞðB2 þ B1 Þ þ pB21 ; 2pB21
ð1 þ aðp þ 1ÞÞ½ð1 þ aðp þ 1ÞÞB2 þ pB21 ; 2pB21 2lpB1 ðpa þ 2a þ 1Þ pB1 B2 ; v :¼ 2 1 þ aðp þ 1Þ B1 ð1 þ aðp þ 1ÞÞ
r3 :¼
If f(z) given by (1) belongs to 8 > < cv 2 apþ2 lapþ1 6 c > : cv
c :¼
pB1 : 2ð1 þ aðp þ 2ÞÞ
S p ða; uÞ, then if l 6 r1 ; if r1 6 l 6 r2 ; if l P r2 :
ð19Þ
Further, if r1 6 l 6 r3, then ð1 þ aðp þ 1ÞÞ2 c ðv þ 1Þjapþ1 j2 6 c: apþ2 la2pþ1 þ p2 B21 If r3 6 l 6 r2, then ð1 þ aðp þ 1ÞÞ2 c ð1 vÞjapþ1 j2 6 c: apþ2 la2pþ1 þ p2 B21 For any complex number l, apþ2 la2pþ1 6 c maxf1; jvjg:
ð20Þ
Further, pB1 H ðq1 ; q2 Þ; að5 p p2 Þ þ 3 where H(q1, q2) is as defined in Lemma 3, 2B2 B21 pðað3p þ 5Þ þ 3Þ q1 :¼ þ 2ðaðp þ 1Þ þ 1Þðaðp þ 2Þ þ 1Þ B1 and B3 B2 pðað3p þ 5Þ þ 3Þ þ p2 B21 ðpa þ a þ 1Þ q2 :¼ þ : 2ðaðp þ 1Þ þ 1Þðaðp þ 2Þ þ 1Þ B1 These results are sharp. japþ3 j 6
ð21Þ
Proof. If f ðzÞ 2 S p ða; uÞ. It is easily shown that 1 þ að1 pÞ zf 0 ðzÞ a z2 f 00 ðzÞ ð1 þ aðp þ 1ÞÞ ð1 þ aðp þ 2ÞÞ ð1 þ aðp þ 1ÞÞ 2 þ :¼ 1 þ apþ1 z þ 2apþ2 apþ1 z2 p f ðzÞ p f ðzÞ p p p 2 ð3 þ að5 p p ÞÞ ð3 þ að3p þ 5ÞÞ ð1 þ aðp þ 1ÞÞ 3 þ apþ3 apþ1 apþ2 þ apþ1 z3 p p p þ
The proof can now be completed as in the proof of Theorem 1. h
ð22Þ
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For the class S p;g ða; uÞ, we have the following result. Theorem 6. Let u(z) = 1 + B1z + B2z2 + B3z3 + , and let r1 :¼ r2 :¼ r3 :¼ v :¼
g2pþ1 ð1 þ aðp þ 1ÞÞ½ð1 þ aðp þ 1ÞÞðB2 B1 Þ þ pB21 2pB21 gpþ2 g2pþ1 ð1 þ aðp þ 1ÞÞ½ð1 þ aðp þ 1ÞÞðB2 þ B1 Þ þ pB21 2pB21 gpþ2 g2pþ1 ð1 þ aðp þ 1ÞÞ½ð1 þ aðp þ 1ÞÞB2 þ pB21 2pB21 gpþ2
2gpþ2 lpB1 ðpa þ 2a þ 1Þ g2pþ1 ð1
þ aðp þ 1ÞÞ
If f(z) given by (1) belongs to 8 > < cv 2 apþ2 lapþ1 6 c > : cv
2
; ;
;
pB1 B2 ; 1 þ aðp þ 1Þ B1
c :¼
pB1 : 2gpþ2 ð1 þ aðp þ 2ÞÞ
S p;g ða; uÞ, then if l 6 r1 ; if r1 6 l 6 r2 ; if l P r2 :
Further, if r1 6 l 6 r3, then ð1 þ aðp þ 1ÞÞ2 g2 c pþ1 ðv þ 1Þjapþ1 j2 6 c: apþ2 la2pþ1 þ 2 2 p B1 If r3 6 l 6 r2, then ð1 þ aðp þ 1ÞÞ2 g2 c pþ1 2 2 ð1 vÞjapþ1 j 6 c: apþ2 lapþ1 þ 2 2 p B1 For any complex number l, apþ2 la2pþ1 6 c maxf1; jvjg: Further, japþ3 j 6
pB1 H ðq1 ; q2 Þ; gpþ3 ½að5 p p2 Þ þ 3
where H(q1, q2) is as defined in Lemma 3, 2B2 B21 pðað3p þ 5Þ þ 3Þ þ q1 :¼ 2ðaðp þ 1Þ þ 1Þðaðp þ 2Þ þ 1Þ B1 and
B3 B2 pðað3p þ 5Þ þ 3Þ þ p2 B21 ðpa þ a þ 1Þ q2 :¼ þ : 2ðaðp þ 1Þ þ 1Þðaðp þ 2Þ þ 1Þ B1 These results are sharp. Remark 2. When p = 1 and uðzÞ ¼
1 þ zð1 2bÞ 1z
ða P 0; 0 6 b < 1Þ;
(19) and (20) of Theorem 5 reduces to [4, Theorem 4 and 3, p. 95]. For the class LM p ða; uÞ, we now get the following coefficient bounds:
ð23Þ
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Theorem 7. Let u(z) = 1 + B1z + B2z2 + B3z3 + . Let c3 ðB2 B1 Þ þ c2 B21 ; c1 B21
r1 :¼
c 1 l c 2 B2 ; c3 B1
T 2 :¼
c3 ðB2 þ B1 Þ þ c2 B21 ; c1 B21 2
c1 :¼ 4p2 ðp þ 2 2aÞ; v :¼
r2 :¼
r3 :¼
c2 :¼ ð1 aÞ½2pðp þ 1Þ þ a þ 2ap3 ; T 1 :¼
c3 B2 þ c2 B21 ; c1 B21 2
c3 :¼ 2ðp þ 1 aÞ ;
3ðp2 þ ð1 aÞð3p þ 2ÞÞ ; p3
1a a 3 ½ðp þ 1Þ ½6pðp þ aÞ þ aða þ 1Þ pa½6p3 þ p2 ð1 þ 7aÞ þ 3pð1 þ 3aÞ þ 3a þ : 6 6p p
If f(z) given by (1) belongs to LM p ða; uÞ, then 8 2 p B1 v > if l 6 r1 ; > 2ðpþ22aÞ > > > < 2 p B1 if r1 6 l 6 r2 ; apþ2 la2pþ1 6 2ðpþ22aÞ > > > > > : p2 B1 v if l P r2 :
ð24Þ
2ðpþ22aÞ
Further, if r1 6 l 6 r3, then c p 2 B1 2 : apþ2 la2pþ1 þ 3 ð1 þ vÞjapþ1 j 6 B 1 c1 2ðp þ 2 2aÞ If r3 6 l 6 r2, then c p 2 B1 2 : apþ2 la2pþ1 þ 3 ð1 vÞjapþ1 j 6 B 1 c1 2ðp þ 2 2aÞ For any complex number l, p 2 B1 max f1; jvjg: apþ2 la2pþ1 6 2ðp þ 2 2aÞ
ð25Þ
ð26Þ
ð27Þ
Further, japþ3 j 6
p 2 B1 H ðq1 ; q2 Þ; 3ðp 3a þ 3Þ
where H(q1, q2) is as defined in Lemma 3, q1 :¼
2B2 T 1 p 4 B1 B1 2ðp a þ 1Þðp 2a þ 2Þ
q2 :¼
B3 T 1 p4 ðc3 B2 þ c2 B21 Þ T 2 p6 B21 : B1 2c3 ðp a þ 1Þðp 2a þ 2Þ ðp a þ 1Þ3
and
These results are sharp. Proof. The proof is similar to the proof of Theorem 1. h Remark 3. As special cases, we note that for p = 1, inequalities (24)–(27) in Theorem 7 are those found in [15, Theorems 2.1 and 2.2, p. 3]. Additionally, if a = 1, then inequalities (24)–(26) are the results established in [14, Theorem 2.1, Remark 2.2, p. 3]. Our final result is on the coefficient bounds for Mp(a, u).
R.M. Ali et al. / Applied Mathematics and Computation 187 (2007) 35–46
45
Theorem 8. Let u(z) = 1 + B1z + B2z2 + B3z3 + Let r1 :¼
c1 ðB2 B1 Þ þ c2 B21 ; c3 B21
r2 :¼
c1 ðB2 þ B1 Þ þ c2 B21 ; c3 B21
2
c2 :¼ a þ pðp þ 2aÞ; c1 :¼ pð1 þ paÞ ; ½pðp þ 2aÞð2l 1Þ aB1 B2 : v :¼ c3 B1
r3 :¼
c1 B2 þ c2 B21 ; c3 B21
c3 :¼ 2pðp þ 2aÞ;
If f(z) given by (1) belongs to Mp(a, u), then 8 2 p B1 v > > > 2ðpþ2aÞ if l 6 r1 ; < p2 B1 if r1 6 l 6 r2 ; apþ2 la2pþ1 6 2ðpþ2aÞ > > > 2 : p B1 v if l P r : 2 2ðpþ2aÞ
ð28Þ
Further, if r1 6 l 6 r3, then c p 2 B1 2 : apþ2 la2pþ1 þ 1 ð1 þ vÞjapþ1 j 6 B1 c 3 2ðp þ 2aÞ
ð29Þ
If r3 6 l 6 r2, then c p 2 B1 2 : apþ2 la2pþ1 þ 1 ð1 vÞjapþ1 j 6 B1 c 3 2ðp þ 2aÞ
ð30Þ
For any complex number l, p 2 B1 max f1; jvjg: apþ2 la2pþ1 6 2ðp þ 2aÞ
ð31Þ
Further, japþ3 j 6
p 2 B1 H ðq1 ; q2 Þ; 3ðp þ 3aÞ
where H(q1, q2) is as defined in Lemma 3, 2B2 3ðp2 þ 3pa þ 2aÞ B1 ; þ 2ð1 þ paÞðp þ 2aÞ B1 B3 3ðp2 þ 3pa þ 2aÞ ðp4 þ 5ap3 þ 3p2 að2a þ 1Þ þ pað9a 2Þ þ 2a2 Þ 2 B2 þ q2 :¼ þ B1 : 3 B1 2ð1 þ paÞðp þ 2aÞ 2pð1 þ paÞ ðp þ 2aÞ
q1 :¼
These results are sharp. Proof. For f(z) 2 Mp(a, u), a computation shows that ! ðp2 þ 2pa þ aÞa2pþ1 2ðp þ 2aÞapþ2 2 1 a zf 0 ðzÞ a zf 00 ðzÞ ð1 þ paÞapþ1 þ ¼1þ zþ z þ 1þ 0 p p3 p2 p f ðzÞ p f ðzÞ 3ðp þ 3aÞ 3ðp2 þ 3pa þ 2aÞ p3 þ 3p2 a þ 3pa þ a 3 þ apþ3 apþ1 apþ2 þ apþ1 z3 p2 p3 p4 þ
The remaining part of the proof is similar to the proof of Theorem 1. b
h
1þz Remark 4. When p = 1 and uðzÞ ¼ ð1z Þ ða P 0; 0 < b 6 1Þ; (28) and (31) of Theorem 8 reduce to [2, Theorems 2.1 and 2.2, p. 23]. When p = 1 and a = 1, (28)–(30) of Theorem 8 reduce to [9, Theorem 3 and Remark, p. 7].
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R.M. Ali et al. / Applied Mathematics and Computation 187 (2007) 35–46
Acknowledgement The authors gratefully acknowledged support from the research grant IRPA 09-02-05-00020 EAR. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
S. Abdul Halim, On a class of functions of complex order, Tamkang J. Math. 30 (2) (1999) 147–153. N.E. Cho, S. Owa, On the Fekete-Szego¨ problem for strongly a-quasiconvex functions, Tamkang J. Math. 34 (1) (2003) 21–28. K.K. Dixit, S.K. Pal, On a class of univalent functions related to complex order, Indian J. Pure Appl. Math. 26 (9) (1995) 889–896. A. Janteng, M. Darus, Coefficient problems for certain classes of analytic functions, J. Inst. Math. Comput. Sci. Math. Ser. 13 (1) (2000) 91–96. F.R. Keogh, E.P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20 (1969) 8–12. K. Kiepiela, M. Pietrzyk, J. Szynal, The sharp bound for some coefficient functional within the class of holomorphic bounded functions and its applications, Rocky Mountain J. Math. 31 (1) (2001) 313–326. O.S. Kwon, N.E. Cho, On the Fekete-Szego¨ problem for certain analytic functions, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 10 (4) (2003) 265–271. J.-L. Li, S. Owa, Sufficient conditions for starlikeness, Indian J. Pure Appl. Math. 33 (3) (2002) 313–318. W.C. Ma, D. Minda, A unified treatment of some special classes of univalent functions, in: Proceedings of the Conference on Complex Analysis (Tianjin, 1992), Internat. Press, Cambridge, MA, pp. 157–169. K.S. Padmanabhan, On sufficient conditions for starlikeness, Indian J. Pure Appl. Math. 32 (4) (2001) 543–550. D.V. Prokhorov, J. Szynal, Inverse coefficients for (a, b)-convex functions, Ann. Univ. Mariae Curie-Skłodowska Sec. A 35 (1981) 125–143, 1984. C. Ramesha, S. Kumar, K.S. Padmanabhan, A sufficient condition for starlikeness, Chinese J. Math. 23 (2) (1995) 167–171. V. Ravichandran, Y. Polatoglu, M. Bolcal, A. Sen, Certain subclasses of starlike and convex functions of complex order, Hacettepe J. Math. Stat. 34 (2005) 9–15. V. Ravichandran, A. Gangadharan, M. Darus, Fekete-Szego¨ inequality for certain class of Bazilevic functions, Far East J. Math. Sci. 15 (2) (2004) 171–180. V. Ravichandran, M. Darus, M. Hussain Khan, K.G. Subramanian, Differential subordination associated with linear operators defined for multivalent functions, Acta Math. Vietnam. 30 (2) (2005) 113–121. V. Ravichandran, Certain applications of first order differential subordination, Far East J. Math. Sci. 12 (1) (2004) 41–51. V. Ravichandran, C. Selvaraj, R. Rajalaksmi, Sufficient conditions for starlike functions of order a, J. Inequal. Pure Appl. Math. 3 (5) (2002). Article 81, 6 pp. (electronic). D.G. Yang, The Fekete-Szego¨ problem for Bazilevicˇ functions of type a and order b, J. Math. Res. Expos. 18 (1) (1998) 99–104.
Coefficient bounds for p-valent functions Rosihan M. Ali *, V. Ravichandran, N. Seenivasagan School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia
Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday
Abstract Sharp bounds for japþ2 la2pþ1 j and jap+3j are derived for certain p-valent analytic functions. These are applied to obtain Fekete-Szego¨ like inequalities for several classes of functions defined by convolution. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Analytic functions; Starlike functions; Convex functions; p-Valent functions; Subordination; Convolution; Fekete-Szego¨ inequalities
1. Introduction Let Ap be the class of all functions of the form 1 X an z n f ðzÞ ¼ zp þ
ð1Þ
n¼pþ1
which are analytic P1in the open unit disk D :¼ {z : jzj < 1} and let A :¼ A1 . For f(z) given by (1) and g(z) given by gðzÞ ¼ zp þ n¼pþ1 bn zn , their convolution (or Hadamard product), denoted by f * g, is defined by 1 X an bn z n : ðf gÞðzÞ :¼ zp þ n¼pþ1
The function f(z) is subordinate to the function g(z), written f(z) g(z), provided there is an analytic function w(z) defined on D with w(0) = 0 and jw(z)j < 1 such that f(z) = g(w(z)). Let u be an analytic function with positive real part in the unit disk D with u(0) = 1 and u 0 (0) > 0 that maps D onto a region starlike with respect to 1 and symmetric with respect to the real axis. We define the class S b;p ðuÞ to be the subclass of Ap consisting of functions f(z) satisfying
*
Corresponding author. E-mail addresses: [email protected] (R.M. Ali), [email protected] (V. Ravichandran), [email protected] (N. Seenivasagan).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.08.100
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R.M. Ali et al. / Applied Mathematics and Computation 187 (2007) 35–46
1 1 zf 0 ðzÞ 1 uðzÞ; 1þ b p f ðzÞ As special cases, let S p ðuÞ :¼ S 1;p ðuÞ;
ðz 2 D and b 2 C n f0gÞ:
S b ðuÞ :¼ S b;1 ðuÞ;
S ðuÞ :¼ S 1;1 ðuÞ:
For a fixed analytic function g 2 Ap with positive coefficients, define the class S b;p;g ðuÞ to be the class of all functions f 2 Ap satisfying f g 2 S b;p ðuÞ. ThisPclass includes as special cases several other classes studied 1 in the literature. For example, when gðzÞ ¼ zp þ n¼pþ1 np zn , the class S b;p;g ðuÞ reduces to the class Cb,p(u) consisting of functions f 2 Ap satisfying 1 1 zf 00 ðzÞ 1þ 0 1 þ uðzÞ; ðz 2 D and b 2 C n f0gÞ: b bp f ðzÞ The classes S*(u) and C(u) :¼ C1,1(u) were introduced and studied by Ma and Minda [9]. Define the class Rb,p(u) to be the class of all functions f 2 Ap satisfying 1 f 0 ðzÞ 1þ 1 uðzÞ; ðz 2 D and b 2 C n f0gÞ; b pzp1 and for a fixed function g with positive coefficients, let Rb,p,g(u) be the class of all functions f 2 Ap satisfying f * g 2 Rb,p(u). Several authors [4,8,10,16,17,12] have studied the classes of analytic functions defined by using the expres0 2 00 ðzÞ sion zff ðzÞ þ a z ffðzÞðzÞ. We shall also consider a class defined by the corresponding quantity for p-valent functions. Define the class S p ða; uÞ to be the class of all functions f 2 Ap satisfying 1 þ að1 pÞ zf 0 ðzÞ a z2 f 00 ðzÞ þ uðzÞ p f ðzÞ p f ðzÞ
ðz 2 D and a P 0Þ:
Note that S p ð0; uÞ is the class S p ðuÞ. Let S p;g ða; uÞ be the class of all functions f 2 Ap for which f g 2 S p ða; uÞ. We shall also consider the class LM p ða; uÞ consisting of p-valent a-convex functions with respect to u. These are functions f 2 Ap satisfying 1 a zf 0 ðzÞ a zf 00 ðzÞ þ 1þ 0 uðzÞ ðz 2 D and a P 0Þ: p f ðzÞ p f ðzÞ Further let Mp(a, u) be the class of functions f 2 Ap satisfying a 1a 1 zf 0 ðzÞ zf 00 ðzÞ 1þ 0 uðzÞ ðz 2 D and a P 0Þ: p f ðzÞ f ðzÞ Functions in this class are called logarithmic p-valent a-convex functions with respect to u. In this paper, we obtain Fekete-Szego¨ inequalities and bounds for the coefficient ap+3 for the classes S p ðuÞ and S p;g ðuÞ. These results are then extended to the other classes defined earlier. See [1–7,9,13,14,18] for FeketeSzego¨ problem for certain related classes of functions. Let X be the class of analytic functions of the form wðzÞ ¼ w1 z þ w2 z2 þ
ð2Þ
in the unit disk D satisfying the condition jw(z)j < 1. We need the following lemmas to prove our main results. Lemma 1. If w 2 X, then 8 > < t if t 6 1; 2 jw2 tw1 j 6 1 if 1 6 t 6 1; > : t if t P 1: When t < 1 or t > 1, equality holds if and only if w(z) = z or one of its rotations. If 1 < t < 1, then equality holds if and only if w(z) = z2 or one of its rotations. Equality holds for t = 1 if and only if wðzÞ ¼
R.M. Ali et al. / Applied Mathematics and Computation 187 (2007) 35–46
37
kþz kþz z 1þkz ð0 6 k 6 1Þ or one of its rotations while for t = 1, equality holds if and only if wðzÞ ¼ z 1þkz ð0 6 k 6 1Þ or one of its rotations. Also the sharp upper bound above can be improved as follows when 1 < t < 1:
jw2 tw21 j þ ðt þ 1Þjw1 j2 6 1 ð1 < t 6 0Þ and jw2 tw21 j þ ð1 tÞjw1 j2 6 1 ð0 < t < 1Þ: Lemma 1 is a reformulation of a Lemma of Ma and Minda [9]. Lemma 2 [5, Inequality 7, p. 10]. If w 2 X, then, for any complex number t, jw2 tw21 j 6 maxf1; jtjg: The result is sharp for the functions w(z) = z2 or w(z) = z. Lemma 3 [11]. If w 2 X, then for any real numbers q1 and q2, the following sharp estimate holds: jw3 þ q1 w1 w2 þ q2 w31 j 6 H ðq1 ; q2 Þ;
ð3Þ
where 8 1 > > > > > > > > > jq2 j > > > > > > > 12 < jq1 jþ1 2 H ðq1 ; q2 Þ ¼ 3 ðjq1 j þ 1Þ 3ðjq1 jþ1þq2 Þ > > > > 2 2 12 > > q1 4 q1 4 1 > > q > 3 2 q21 4q2 3ðq2 1Þ > > > > > 12 > > jq1 j1 : 2 ðjq j 1Þ 3
1
3ðjq1 j1q2 Þ
for ðq1 ; q2 Þ 2 D1 [ D2 ; for ðq1 ; q2 Þ 2
7 S
Dk ;
k¼3
for ðq1 ; q2 Þ 2 D8 [ D9 ; for ðq1 ; q2 Þ 2 D10 [ D11 f2; 1g; for ðq1 ; q2 Þ 2 D12 :
The extremal functions, up to rotations, are of the form wðzÞ ¼ z3 ;
wðzÞ ¼ z;
wðzÞ ¼ w1 ðzÞ ¼ je1 j ¼ je2 j ¼ 1;
zðt1 zÞ ; 1 t1 z
wðzÞ ¼ w0 ðzÞ ¼
zð½ð1 kÞe2 þ ke1 e1 e2 zÞ ; 1 ½ð1 kÞe1 þ ke2 z
wðzÞ ¼ w2 ðzÞ ¼ ih0
zðt2 þ zÞ ; 1 þ t2 z ih0
e1 ¼ t0 e 2 ða bÞ; e2 ¼ e 2 ðia bÞ; rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h0 h0 ba ; b ¼ 1 t20 sin2 ; k¼ a ¼ t0 cos ; 2b 2 2 1 12 2q2 ðq21 þ 2Þ 3q21 2 jq1 j þ 1 t0 ¼ ; t1 ¼ ; 3ðjq1 j þ 1 þ q2 Þ 3ðq2 1Þðq21 4q2 Þ 12 jq1 j 1 h0 q1 q2 ðq21 þ 8Þ 2ðq21 þ 2Þ t2 ¼ ; cos ¼ : 3ðjq1 j 1 q2 Þ 2 2 2q2 ðq21 þ 2Þ 3q21
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R.M. Ali et al. / Applied Mathematics and Computation 187 (2007) 35–46
The sets Dk, k = 1, 2, . . . , 12, are defined as follows:
1 D1 ¼ ðq1 ; q2 Þ : jq1 j 6 ; jq2 j 6 1 ; 2
1 4 D2 ¼ ðq1 ; q2 Þ : 6 jq1 j 6 2; ðjq1 j þ 1Þ3 ðjq1 j þ 1Þ 6 q2 6 1 ; 2 27
1 D3 ¼ ðq1 ; q2 Þ : jq1 j 6 ; q2 6 1 ; 2
1 2 D4 ¼ ðq1 ; q2 Þ : jq1 j P ; q2 6 ðjq1 j þ 1Þ ; 2 3 D5 ¼ fðq1 ; q2 Þ : jq1 j 6 2; q2 P 1g;
1 D6 ¼ ðq1 ; q2 Þ : 2 6 jq1 j 6 4; q2 P ðq21 þ 8Þ ; 12
2 D7 ¼ ðq1 ; q2 Þ : jq1 j P 4; q2 P ðjq1 j 1Þ ; 3
1 2 4 D8 ¼ ðq1 ; q2 Þ : 6 jq1 j 6 2; ðjq1 j þ 1Þ 6 q2 6 ðjq1 j þ 1Þ3 ðjq1 j þ 1Þ ; 2 3 27
2 2jq jðjq j þ 1Þ D9 ¼ ðq1 ; q2 Þ : jq1 j P 2; ðjq1 j þ 1Þ 6 q2 6 2 1 1 ; 3 q1 þ 2jq1 j þ 4
2jq1 jðjq1 j þ 1Þ 1 2 6 q2 6 ðq1 þ 8Þ ; D10 ¼ ðq1 ; q2 Þ : 2 6 jq1 j 6 4; 2 q1 þ 2jq1 j þ 4 12
2jq jðjq j þ 1Þ 2jq jðjq j 1Þ 6 q2 6 2 1 1 D11 ¼ ðq1 ; q2 Þ : jq1 j P 4; 2 1 1 ; q1 þ 2jq1 j þ 4 q1 2jq1 j þ 4
2jq jðjq j 1Þ 2 6 q2 6 ðjq1 j 1Þ : D12 ¼ ðq1 ; q2 Þ : jq1 j P 4; 2 1 1 q1 2jq1 j þ 4 3 2. Coefficient bounds By making use of the Lemmas 1–3, we prove the following bounds for the class S p;g ðuÞ: Theorem 1. Let u(z) = 1 + B1z + B2z2 + B3z3 + , and r1 :¼
B2 B1 þ pB21 ; 2pB21
r2 :¼
B2 þ B1 þ pB21 ; 2pB21
If f(z) given by (1) belongs to S p ðuÞ, then
8p 2 > < 2 B2 þ ð1 2lÞpB1 apþ2 la2pþ1 6 pB2 1 >
: p 2 B2 þ ð1 2lÞpB21
r3 :¼
B2 þ pB21 : 2pB21
if l 6 r1 ; if r1 6 l 6 r2 ; if l P r2 :
Further, if r1 6 l 6 r3, then 1 B2 pB 2 2 1 þ ð2l 1ÞpB1 japþ1 j 6 1 : apþ2 lapþ1 þ 2pB1 B1 2 If r3 6 l 6 r2, then 1 B2 pB 2 2 1 þ ð2l 1ÞpB1 japþ1 j 6 1 : apþ2 lapþ1 þ 2pB1 B1 2
ð4Þ
ð5Þ
ð6Þ
R.M. Ali et al. / Applied Mathematics and Computation 187 (2007) 35–46
For any complex number l,
pB B2 apþ2 la2pþ1 6 1 max 1; þ ð1 2lÞpB1 : 2 B1 Further, pB japþ3 j 6 1 H ðq1 ; q2 Þ; 3 where H(q1, q2) is as defined in Lemma 3, q1 :¼
4B2 þ 3pB21 2B1
and
q2 :¼
39
ð7Þ
ð8Þ
2B3 þ 3pB1 B2 þ p2 B31 : 2B1
These results are sharp. Proof. If f ðzÞ 2 S p ðuÞ; then there is an analytic function w(z) = w1z + w2z2 + 2 X such that zf 0 ðzÞ ¼ uðwðzÞÞ: pf ðzÞ Since zf 0 ðzÞ apþ1 ¼1þ zþ pf ðzÞ p
ð9Þ ! a2pþ1 2apþ2 2 þ z þ p p
! a3pþ1 3 3apþ3 3 apþ1 apþ2 þ z þ ; p p p
we have from (9), apþ1 ¼ pB1 w1 ; 1 pB1 w2 þ pðB2 þ pB21 Þw21 apþ2 ¼ 2 and apþ3
ð10Þ ð11Þ
pB1 4B2 þ 3pB21 2B3 þ 3pB1 B2 þ p2 B31 3 w3 þ ¼ w1 w2 þ w1 : 3 2B1 2B1
ð12Þ
Using (10) and (11), we have pB ð13Þ apþ2 la2pþ1 ¼ 1 w2 vw21 ; 2 where B2 v :¼ pB1 ð2l 1Þ : B1 The results (4)–(6) are established by an application of Lemma 1, inequality (7) by Lemma 2 and (8) follows from Lemma 3. To show that the bounds in (4)–(6) are sharp, we define the functions Kun (n = 2, 3, . . .) by zK 0un ðzÞ ¼ uðzn1 Þ; pK un ðzÞ
0
K un ð0Þ ¼ 0 ¼ ½K un ð0Þ 1
and the functions Fk and Gk (0 6 k 6 1) by zF 0k ðzÞ zðz þ kÞ ¼u ; F k ð0Þ ¼ 0 ¼ F 0k ð0Þ 1 pF k ðzÞ 1 þ kz and zG0k ðzÞ zðz þ kÞ ¼u ; Gk ð0Þ ¼ 0 ¼ G0k ð0Þ 1: pGk ðzÞ 1 þ kz Clearly the functions K un ; F k ; Gk 2 S p ðuÞ. We shall also write Ku :¼ Ku2.
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R.M. Ali et al. / Applied Mathematics and Computation 187 (2007) 35–46
If l < r1 or l > r2, then equality holds if and only if f is Ku or one of its rotations. When r1 < l < r2, then equality holds if and only if f is Ku3 or one of its rotations. If l = r1 then equality holds if and only if f is Fk or one of its rotations. Equality holds for l = r2 if and only if f is Gk or one of its rotations. h Corollary 1. Let u(z) = 1 + B1z + B2z2 + B3z3 + , and let r1 :¼
g2pþ1 B2 B1 þ pB21 ; gpþ2 2pB21
r2 :¼
g2pþ1 B2 þ B1 þ pB21 ; gpþ2 2pB21
If f(z) given by (1) belongs to S p;g ðuÞ, then 8 gpþ2 > p 2 > > > 2gpþ2 B2 þ 1 2 g2pþ1 l B1 > > < pB1 apþ2 la2pþ1 6 2gpþ2 > > > > gpþ2 > p > B þ 1 2 l B21 : 2gpþ2 2 g2
r3 :¼
g2pþ1 B2 þ pB21 : gpþ2 2pB21
if l 6 r1 ; if r1 6 l 6 r2 ; if l P r2 :
pþ1
Further, if r1 6 l 6 r3, then apþ2 la2pþ1 þ
g2pþ1 2gpþ2 pB1
B2 1 þ B1
! ! gpþ2 pB1 2 2 2 l 1 B1 japþ1 j 6 : gpþ1 2gpþ2
If r3 6 l 6 r2, then apþ2 la2pþ1 þ
g2pþ1 2gpþ2 pB1
B2 1þ B1
! ! gpþ2 pB1 2 2 2 l 1 B1 japþ1 j 6 : gpþ1 2gpþ2
For any complex number l, ( B pB 2 1 max 1; þ apþ2 la2pþ1 6 B1 2gpþ2
! ) gpþ2 1 2 2 l B1 : gpþ1
Further, japþ3 j 6
pB1 H ðq1 ; q2 Þ; 3gpþ3
ð14Þ
where H(q1, q2) is as defined in Lemma 3, q1 :¼
4B2 þ 3pB21 2B1
and
q2 :¼
2B3 þ 3pB1 B2 þ p2 B31 : 2B1
These results are sharp. Theorem 2. Let u be as in Theorem 1. If f(z) given by (1) belongs to S b;p;g ðuÞ, then, for any complex number l, we have ( ! ) B pjbjB g2pþ1 2 1 2 max 1; þ bp 1 2 l B1 : apþ2 lapþ1 6 B1 2gpþ2 gpþ2 The result is sharp. Proof. The proof is similar to the proof of Theorem 1. h Proceeding similarly, we now obtain coefficient bounds for the class Rb,p,g(u).
R.M. Ali et al. / Applied Mathematics and Computation 187 (2007) 35–46
41
Theorem 3. Let u(z) = 1 + B1z + B2z2 + B3z3 + If f(z) given by (1) belongs to Rb,p(u), then for any complex number l, ð15Þ apþ2 la2pþ1 6 jcj max f1; jvjg; where v :¼ l
pbB1 ðp þ 2Þ ðp þ 1Þ
2
B2 B1
and
c :¼
bpB1 : 2þp
Further, japþ3 j 6
jbjpB1 H ðq1 ; q2 Þ; 3þp
ð16Þ
where H(q1, q2) is as defined in Lemma 3, q1 :¼
2B2 B1
and
q2 :¼
B3 : B1
These results are sharp. Proof. A computation shows that 1 f 0 ðzÞ pþ1 pþ2 pþ3 apþ1 z þ apþ2 z2 þ apþ3 z3 þ : 1þ 1 ¼1þ b pzp1 bp bp bp Thus bpB1 w2 vw21 ¼ c w2 vw21 ; apþ2 la2pþ1 ¼ pþ2 h i 1 lð2þpÞ 1 where v :¼ bpBðpþ1Þ BB21 and c :¼ bpB . The result now follows from Lemmas 2 and 3. 2 pþ2
ð17Þ h
Remark 1. When p = 1 and 1 þ Az ð1 6 B 6 A 6 1Þ; uðzÞ ¼ 1 þ Bz inequality (15) reduces to give the inequality [3, Theorem 4, p. 894]. For the class Rb,p,g(u), we have the following result. Theorem 4. Let u(z) = 1 + B1z + B2z2 + B3z3 + . If f(z) given by (1) belongs to Rb,p(u), then for any complex number l, apþ2 la2pþ1 6 jcj max f1; jvjg; where v :¼
lg2pþ1 pbB1 ðp þ 2Þ B2 gpþ2 ðp þ 1Þ2 B1
and
c :¼
bpB1 : gpþ2 ð2 þ pÞ
Further, japþ3 j 6
jbjpB1 H ðq1 ; q2 Þ; ð3 þ pÞgpþ3
where H(q1, q2) is as defined in Lemma 3, q1 :¼
2B2 B1
and
These results are sharp.
q2 :¼
B3 : B1
ð18Þ
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We now prove the coefficient bounds for S p;g ða; uÞ. Theorem 5. Let u(z) = 1 + B1z + B2z2 + B3z3 + Further let r1 :¼
ð1 þ aðp þ 1ÞÞ½ð1 þ aðp þ 1ÞÞðB2 B1 Þ þ pB21 ; 2pB21
r2 :¼
ð1 þ aðp þ 1ÞÞ½ð1 þ aðp þ 1ÞÞðB2 þ B1 Þ þ pB21 ; 2pB21
ð1 þ aðp þ 1ÞÞ½ð1 þ aðp þ 1ÞÞB2 þ pB21 ; 2pB21 2lpB1 ðpa þ 2a þ 1Þ pB1 B2 ; v :¼ 2 1 þ aðp þ 1Þ B1 ð1 þ aðp þ 1ÞÞ
r3 :¼
If f(z) given by (1) belongs to 8 > < cv 2 apþ2 lapþ1 6 c > : cv
c :¼
pB1 : 2ð1 þ aðp þ 2ÞÞ
S p ða; uÞ, then if l 6 r1 ; if r1 6 l 6 r2 ; if l P r2 :
ð19Þ
Further, if r1 6 l 6 r3, then ð1 þ aðp þ 1ÞÞ2 c ðv þ 1Þjapþ1 j2 6 c: apþ2 la2pþ1 þ p2 B21 If r3 6 l 6 r2, then ð1 þ aðp þ 1ÞÞ2 c ð1 vÞjapþ1 j2 6 c: apþ2 la2pþ1 þ p2 B21 For any complex number l, apþ2 la2pþ1 6 c maxf1; jvjg:
ð20Þ
Further, pB1 H ðq1 ; q2 Þ; að5 p p2 Þ þ 3 where H(q1, q2) is as defined in Lemma 3, 2B2 B21 pðað3p þ 5Þ þ 3Þ q1 :¼ þ 2ðaðp þ 1Þ þ 1Þðaðp þ 2Þ þ 1Þ B1 and B3 B2 pðað3p þ 5Þ þ 3Þ þ p2 B21 ðpa þ a þ 1Þ q2 :¼ þ : 2ðaðp þ 1Þ þ 1Þðaðp þ 2Þ þ 1Þ B1 These results are sharp. japþ3 j 6
ð21Þ
Proof. If f ðzÞ 2 S p ða; uÞ. It is easily shown that 1 þ að1 pÞ zf 0 ðzÞ a z2 f 00 ðzÞ ð1 þ aðp þ 1ÞÞ ð1 þ aðp þ 2ÞÞ ð1 þ aðp þ 1ÞÞ 2 þ :¼ 1 þ apþ1 z þ 2apþ2 apþ1 z2 p f ðzÞ p f ðzÞ p p p 2 ð3 þ að5 p p ÞÞ ð3 þ að3p þ 5ÞÞ ð1 þ aðp þ 1ÞÞ 3 þ apþ3 apþ1 apþ2 þ apþ1 z3 p p p þ
The proof can now be completed as in the proof of Theorem 1. h
ð22Þ
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43
For the class S p;g ða; uÞ, we have the following result. Theorem 6. Let u(z) = 1 + B1z + B2z2 + B3z3 + , and let r1 :¼ r2 :¼ r3 :¼ v :¼
g2pþ1 ð1 þ aðp þ 1ÞÞ½ð1 þ aðp þ 1ÞÞðB2 B1 Þ þ pB21 2pB21 gpþ2 g2pþ1 ð1 þ aðp þ 1ÞÞ½ð1 þ aðp þ 1ÞÞðB2 þ B1 Þ þ pB21 2pB21 gpþ2 g2pþ1 ð1 þ aðp þ 1ÞÞ½ð1 þ aðp þ 1ÞÞB2 þ pB21 2pB21 gpþ2
2gpþ2 lpB1 ðpa þ 2a þ 1Þ g2pþ1 ð1
þ aðp þ 1ÞÞ
If f(z) given by (1) belongs to 8 > < cv 2 apþ2 lapþ1 6 c > : cv
2
; ;
;
pB1 B2 ; 1 þ aðp þ 1Þ B1
c :¼
pB1 : 2gpþ2 ð1 þ aðp þ 2ÞÞ
S p;g ða; uÞ, then if l 6 r1 ; if r1 6 l 6 r2 ; if l P r2 :
Further, if r1 6 l 6 r3, then ð1 þ aðp þ 1ÞÞ2 g2 c pþ1 ðv þ 1Þjapþ1 j2 6 c: apþ2 la2pþ1 þ 2 2 p B1 If r3 6 l 6 r2, then ð1 þ aðp þ 1ÞÞ2 g2 c pþ1 2 2 ð1 vÞjapþ1 j 6 c: apþ2 lapþ1 þ 2 2 p B1 For any complex number l, apþ2 la2pþ1 6 c maxf1; jvjg: Further, japþ3 j 6
pB1 H ðq1 ; q2 Þ; gpþ3 ½að5 p p2 Þ þ 3
where H(q1, q2) is as defined in Lemma 3, 2B2 B21 pðað3p þ 5Þ þ 3Þ þ q1 :¼ 2ðaðp þ 1Þ þ 1Þðaðp þ 2Þ þ 1Þ B1 and
B3 B2 pðað3p þ 5Þ þ 3Þ þ p2 B21 ðpa þ a þ 1Þ q2 :¼ þ : 2ðaðp þ 1Þ þ 1Þðaðp þ 2Þ þ 1Þ B1 These results are sharp. Remark 2. When p = 1 and uðzÞ ¼
1 þ zð1 2bÞ 1z
ða P 0; 0 6 b < 1Þ;
(19) and (20) of Theorem 5 reduces to [4, Theorem 4 and 3, p. 95]. For the class LM p ða; uÞ, we now get the following coefficient bounds:
ð23Þ
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R.M. Ali et al. / Applied Mathematics and Computation 187 (2007) 35–46
Theorem 7. Let u(z) = 1 + B1z + B2z2 + B3z3 + . Let c3 ðB2 B1 Þ þ c2 B21 ; c1 B21
r1 :¼
c 1 l c 2 B2 ; c3 B1
T 2 :¼
c3 ðB2 þ B1 Þ þ c2 B21 ; c1 B21 2
c1 :¼ 4p2 ðp þ 2 2aÞ; v :¼
r2 :¼
r3 :¼
c2 :¼ ð1 aÞ½2pðp þ 1Þ þ a þ 2ap3 ; T 1 :¼
c3 B2 þ c2 B21 ; c1 B21 2
c3 :¼ 2ðp þ 1 aÞ ;
3ðp2 þ ð1 aÞð3p þ 2ÞÞ ; p3
1a a 3 ½ðp þ 1Þ ½6pðp þ aÞ þ aða þ 1Þ pa½6p3 þ p2 ð1 þ 7aÞ þ 3pð1 þ 3aÞ þ 3a þ : 6 6p p
If f(z) given by (1) belongs to LM p ða; uÞ, then 8 2 p B1 v > if l 6 r1 ; > 2ðpþ22aÞ > > > < 2 p B1 if r1 6 l 6 r2 ; apþ2 la2pþ1 6 2ðpþ22aÞ > > > > > : p2 B1 v if l P r2 :
ð24Þ
2ðpþ22aÞ
Further, if r1 6 l 6 r3, then c p 2 B1 2 : apþ2 la2pþ1 þ 3 ð1 þ vÞjapþ1 j 6 B 1 c1 2ðp þ 2 2aÞ If r3 6 l 6 r2, then c p 2 B1 2 : apþ2 la2pþ1 þ 3 ð1 vÞjapþ1 j 6 B 1 c1 2ðp þ 2 2aÞ For any complex number l, p 2 B1 max f1; jvjg: apþ2 la2pþ1 6 2ðp þ 2 2aÞ
ð25Þ
ð26Þ
ð27Þ
Further, japþ3 j 6
p 2 B1 H ðq1 ; q2 Þ; 3ðp 3a þ 3Þ
where H(q1, q2) is as defined in Lemma 3, q1 :¼
2B2 T 1 p 4 B1 B1 2ðp a þ 1Þðp 2a þ 2Þ
q2 :¼
B3 T 1 p4 ðc3 B2 þ c2 B21 Þ T 2 p6 B21 : B1 2c3 ðp a þ 1Þðp 2a þ 2Þ ðp a þ 1Þ3
and
These results are sharp. Proof. The proof is similar to the proof of Theorem 1. h Remark 3. As special cases, we note that for p = 1, inequalities (24)–(27) in Theorem 7 are those found in [15, Theorems 2.1 and 2.2, p. 3]. Additionally, if a = 1, then inequalities (24)–(26) are the results established in [14, Theorem 2.1, Remark 2.2, p. 3]. Our final result is on the coefficient bounds for Mp(a, u).
R.M. Ali et al. / Applied Mathematics and Computation 187 (2007) 35–46
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Theorem 8. Let u(z) = 1 + B1z + B2z2 + B3z3 + Let r1 :¼
c1 ðB2 B1 Þ þ c2 B21 ; c3 B21
r2 :¼
c1 ðB2 þ B1 Þ þ c2 B21 ; c3 B21
2
c2 :¼ a þ pðp þ 2aÞ; c1 :¼ pð1 þ paÞ ; ½pðp þ 2aÞð2l 1Þ aB1 B2 : v :¼ c3 B1
r3 :¼
c1 B2 þ c2 B21 ; c3 B21
c3 :¼ 2pðp þ 2aÞ;
If f(z) given by (1) belongs to Mp(a, u), then 8 2 p B1 v > > > 2ðpþ2aÞ if l 6 r1 ; < p2 B1 if r1 6 l 6 r2 ; apþ2 la2pþ1 6 2ðpþ2aÞ > > > 2 : p B1 v if l P r : 2 2ðpþ2aÞ
ð28Þ
Further, if r1 6 l 6 r3, then c p 2 B1 2 : apþ2 la2pþ1 þ 1 ð1 þ vÞjapþ1 j 6 B1 c 3 2ðp þ 2aÞ
ð29Þ
If r3 6 l 6 r2, then c p 2 B1 2 : apþ2 la2pþ1 þ 1 ð1 vÞjapþ1 j 6 B1 c 3 2ðp þ 2aÞ
ð30Þ
For any complex number l, p 2 B1 max f1; jvjg: apþ2 la2pþ1 6 2ðp þ 2aÞ
ð31Þ
Further, japþ3 j 6
p 2 B1 H ðq1 ; q2 Þ; 3ðp þ 3aÞ
where H(q1, q2) is as defined in Lemma 3, 2B2 3ðp2 þ 3pa þ 2aÞ B1 ; þ 2ð1 þ paÞðp þ 2aÞ B1 B3 3ðp2 þ 3pa þ 2aÞ ðp4 þ 5ap3 þ 3p2 að2a þ 1Þ þ pað9a 2Þ þ 2a2 Þ 2 B2 þ q2 :¼ þ B1 : 3 B1 2ð1 þ paÞðp þ 2aÞ 2pð1 þ paÞ ðp þ 2aÞ
q1 :¼
These results are sharp. Proof. For f(z) 2 Mp(a, u), a computation shows that ! ðp2 þ 2pa þ aÞa2pþ1 2ðp þ 2aÞapþ2 2 1 a zf 0 ðzÞ a zf 00 ðzÞ ð1 þ paÞapþ1 þ ¼1þ zþ z þ 1þ 0 p p3 p2 p f ðzÞ p f ðzÞ 3ðp þ 3aÞ 3ðp2 þ 3pa þ 2aÞ p3 þ 3p2 a þ 3pa þ a 3 þ apþ3 apþ1 apþ2 þ apþ1 z3 p2 p3 p4 þ
The remaining part of the proof is similar to the proof of Theorem 1. b
h
1þz Remark 4. When p = 1 and uðzÞ ¼ ð1z Þ ða P 0; 0 < b 6 1Þ; (28) and (31) of Theorem 8 reduce to [2, Theorems 2.1 and 2.2, p. 23]. When p = 1 and a = 1, (28)–(30) of Theorem 8 reduce to [9, Theorem 3 and Remark, p. 7].
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Acknowledgement The authors gratefully acknowledged support from the research grant IRPA 09-02-05-00020 EAR. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
S. Abdul Halim, On a class of functions of complex order, Tamkang J. Math. 30 (2) (1999) 147–153. N.E. Cho, S. Owa, On the Fekete-Szego¨ problem for strongly a-quasiconvex functions, Tamkang J. Math. 34 (1) (2003) 21–28. K.K. Dixit, S.K. Pal, On a class of univalent functions related to complex order, Indian J. Pure Appl. Math. 26 (9) (1995) 889–896. A. Janteng, M. Darus, Coefficient problems for certain classes of analytic functions, J. Inst. Math. Comput. Sci. Math. Ser. 13 (1) (2000) 91–96. F.R. Keogh, E.P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20 (1969) 8–12. K. Kiepiela, M. Pietrzyk, J. Szynal, The sharp bound for some coefficient functional within the class of holomorphic bounded functions and its applications, Rocky Mountain J. Math. 31 (1) (2001) 313–326. O.S. Kwon, N.E. Cho, On the Fekete-Szego¨ problem for certain analytic functions, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 10 (4) (2003) 265–271. J.-L. Li, S. Owa, Sufficient conditions for starlikeness, Indian J. Pure Appl. Math. 33 (3) (2002) 313–318. W.C. Ma, D. Minda, A unified treatment of some special classes of univalent functions, in: Proceedings of the Conference on Complex Analysis (Tianjin, 1992), Internat. Press, Cambridge, MA, pp. 157–169. K.S. Padmanabhan, On sufficient conditions for starlikeness, Indian J. Pure Appl. Math. 32 (4) (2001) 543–550. D.V. Prokhorov, J. Szynal, Inverse coefficients for (a, b)-convex functions, Ann. Univ. Mariae Curie-Skłodowska Sec. A 35 (1981) 125–143, 1984. C. Ramesha, S. Kumar, K.S. Padmanabhan, A sufficient condition for starlikeness, Chinese J. Math. 23 (2) (1995) 167–171. V. Ravichandran, Y. Polatoglu, M. Bolcal, A. Sen, Certain subclasses of starlike and convex functions of complex order, Hacettepe J. Math. Stat. 34 (2005) 9–15. V. Ravichandran, A. Gangadharan, M. Darus, Fekete-Szego¨ inequality for certain class of Bazilevic functions, Far East J. Math. Sci. 15 (2) (2004) 171–180. V. Ravichandran, M. Darus, M. Hussain Khan, K.G. Subramanian, Differential subordination associated with linear operators defined for multivalent functions, Acta Math. Vietnam. 30 (2) (2005) 113–121. V. Ravichandran, Certain applications of first order differential subordination, Far East J. Math. Sci. 12 (1) (2004) 41–51. V. Ravichandran, C. Selvaraj, R. Rajalaksmi, Sufficient conditions for starlike functions of order a, J. Inequal. Pure Appl. Math. 3 (5) (2002). Article 81, 6 pp. (electronic). D.G. Yang, The Fekete-Szego¨ problem for Bazilevicˇ functions of type a and order b, J. Math. Res. Expos. 18 (1) (1998) 99–104.
