Majorization properties for certain new classes of analytic functions ...
Li et al. Journal of Inequalities and Applications 2013, 2013:86 http://www.journalofinequalitiesandapplications.com/content/2013/1/86
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Majorization properties for certain new classes of analytic functions using the Salagean operator Shu-Hai Li1* , Huo Tang2,1 and En Ao1 *
Correspondence: [email protected] 1 School of Mathematics and Statistics, Chifeng University, Chifeng, Inner Mongolia 024000, China Full list of author information is available at the end of the article
Abstract In the present paper, we investigate the majorization properties for certain classes of multivalent analytic functions defined by the Salagean operator. Moreover, we point out some new and interesting consequences of our main result. MSC: 30C45 Keywords: analytic functions; multivalent functions; α -uniformly starlike functions of order β ; α -uniformly convex functions of order β ; subordination; majorization property
1 Introduction and definitions Let f and g be two analytic functions in the open unit disk = z ∈ C : |z| < .
(.)
We say that f is majorized by g in (see []) and write f (z) g(z)
(z ∈ )
(.)
if there exists a function ϕ, analytic in , such that ϕ(z) ≤ and
f (z) = ϕ(z)g(z)
(z ∈ ).
(.)
It may be noted here that (.) is closely related to the concept of quasi-subordination between analytic functions. For two functions f and g, analytic in , we say that the function f is subordinate to g in if there exists a Schwarz function ω, which is analytic in with ω() = and
ω(z) <
(z ∈ ),
such that f (z) = g ω(z)
(z ∈ ).
© 2013 Li et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Li et al. Journal of Inequalities and Applications 2013, 2013:86 http://www.journalofinequalitiesandapplications.com/content/2013/1/86
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We denote this subordination by f (z) ≺ g(z). Furthermore, if the function g is univalent in , then f (z) ≺ g(z)
(z ∈ )
⇔
f () = g() and
f () ⊂ g().
Let Ap denote the class of functions of the form ∞
f (z) = zp +
ak z k
p ∈ N = {, , . . .} ,
(.)
k=p+
that are analytic and p-valent in the open unit disk . Also, let A = A. For a function f ∈ Ap , let f (q) denote a qth-order ordinary differential operator by
f
(q)
∞ p! k! p–q z + ak zk–q , (z) = (p – q)! (k – q)!
(.)
k=p+
where p > q, p ∈ N , q ∈ N = N ∪ {} and z ∈ . Next, Frasin [] introduced the differential operator Dm f (q) as follows: Dm f (q) (z) =
∞ p!(p – q)m p–q k!(k – q)m z + ak zk–q . (p – q)! (k – q)!
(.)
k=p+
In view of (.), it is clear that D f () (z) = f (z), D f () (z) = zf (z) and Dm f () (z) = Dm f (z) is a known operator introduced by Salagean []. j,l
Definition . A function f (z) ∈ Ap is said to be in the class Lp,q [A, B; α, γ ] of p-valent functions of complex order γ = in if and only if
j (q)
D f (z) + Az Dj f (q) (z) j–l j–l – α ≺ – (p – q) – (p – q) l (q) l (q) D f (z) γ D f (z) + Bz ∗ z ∈ ; – ≤ B < A ≤ ; j > l; p, j ∈ N; l, q ∈ N ; ≤ α; γ ∈ C = C \ {} .
+
γ
(.)
Clearly, we have the following relationships: j,l j,l () Lp,q [A, B; , γ ] = Sp,q [A, B; γ ]; m,n () L, [A, B; α, ] = Um,n (α, A, B); () L, , [ – β, –; α, ] = US(α, β) ( ≤ β < ) (α-uniformly starlike functions of order β); () L, , [ – β, –; α, ] = UK(α, β) ( ≤ β < ) (α-uniformly convex functions of order β); () Ln+,n p, [, –; α, γ ] = Sn (p, α, γ ) (n ∈ N ); ∗ () L, , [, –; α, γ ] = S(α, γ ) ( ≤ α < , γ ∈ C ); , () L, [, –; α, γ ] = K(α, γ ) ( ≤ α < , γ ∈ C ∗ ); ∗ () L, , [, –; α, – β] = S (α, β) ( ≤ α < , ≤ β < ). j,l
The classes Sp,q [A, B; γ ] and Um,n (α, A, B) were introduced by Goswami and Aouf [] and Li and Tang [], respectively. The classes US(α, β) and UK(α, β) were studied recently
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in [] (see also [–]). The class Sn (p, , γ ) = Sn (p, γ ) was introduced by Akbulut et al. []. Also, the classes S(, γ ) = S(γ ) and K(, γ ) = K(γ ) are said to be classes of starlike and convex of complex order γ = in which were considered by Nasr and Aouf [] and Wiatrowski [] (see also [, ]), and S∗ (, β) = S∗ (β) denotes the class of starlike functions of order β in . A majorization problem for the class S(γ ) has recently been investigated by Altintas j,l et al. []. Also, majorization problems for the classes S∗ (β) and Sp,q [A, B; γ ] have been investigated by MacGregor [] and Goswami and Aouf [], respectively. Very recently, Goyal and Goswami [] (see also []) generalized these results for the fractional derivative operator. In the present paper, we investigate a majorization problem for the class j,l Lp,q [A, B; α, γ ]. j,l
2 Majorization problem for the class Lp,q [A, B; α , γ ] We begin by proving the following result. j,l
Theorem . Let the function f ∈ Ap and suppose that g ∈ Lp,q [A, B; α, γ ]. If Dj f (q) (z) is majorized by Dl g (q) (z) in , and (p – q)j–l ≥
(A – B)|γ | + (p – q)j–l |B| δ, –α
then j+ (q) l+ (q) D f (z) ≤ D g (z)
|z| ≤ r ,
(.)
where r = r (p, q, α, γ , j, l, A, B) is the smallest positive root of the equation
(A – B)|γ | + (p – q)j–l |B| r – (p – q)j–l + |B| r –α (A – B)|γ | + (p – q)j–l |B| + r + (p – q)j–l = – –α – ≤ B < A ≤ ; p, j ∈ N; q, l ∈ N ; ≤ α < ; γ ∈ C ∗ , ≤ δ ≤ r . j,l
Proof Suppose that g ∈ Lp,q [A, B; α, γ ]. Then, making use of the fact that – α| – | ≺
+ Az + Bz
⇔
+ Az – αe–iφ + αe–iφ ≺ + Bz
and letting =+ γ
Dj g (q) (z) – (p – q)j–l Dl g (q) (z)
in (.), we obtain +
γ
Dj g (q) (z) – (p – q)j–l Dl g (q) (z)
+ Az – αe–iφ + αe–iφ ≺ + Bz
(φ ∈ R),
(.)
Li et al. Journal of Inequalities and Applications 2013, 2013:86 http://www.journalofinequalitiesandapplications.com/content/2013/1/86
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or, equivalently, + γ
–iφ
+ ( A–αBe )z Dj g (q) (z) –αe–iφ j–l ≺ – (p – q) l (q) D g (z) + Bz
(.)
which holds true for all z ∈ . We find from (.) that + γ
–iφ
+ ( A–αBe )ω(z) Dj g (q) (z) –αe–iφ j–l = – (p – q) , Dl g (q) (z) + Bω(z)
(.)
where ω(z) = c z+c z +· · · , ω ∈ P, P denotes the well-known class of the bounded analytic functions in and satisfies the conditions ω() = and
ω(z) ≤ |z|
(z ∈ ).
From (.), we get (A–B)γ j–l j–l Dj g (q) (z) (p – q) + [ –αe–iφ + (p – q) B]ω(z) = . Dl g (q) (z) + Bω(z)
(.)
By virtue of (.), we obtain l (q) D g (z) ≤ ≤
+ |B||z| (p – q)j–l
(A–B)γ – | –αe –iφ
+ (p – q)j–l B||z|
j (q) D g (z)
+ |B||z| | (p – q)j–l – [ (A–B)|γ –α
+ (p – q)j–l |B|]|z|
j (q) D g (z).
(.)
Next, since Dj f (q) (z) is majorized by Dl g (q) (z) in , thus from (.), we have Dj f (q) (z) = ϕ(z)Dl g (q) (z). Differentiating the above equality with respect to z and multiplying by z, we get Dj+ f (q) (z) = zϕ (z)Dl g (q) (z) + ϕ(z)Dl+ g (q) (z).
(.)
Thus, by noting that ϕ(z) ∈ P satisfies the inequality (see, e.g., Nehari []) – |ϕ(z)| ϕ (z) ≤ – |z|
(z ∈ )
(.)
and making use of (.) and (.) in (.), we obtain
j+ (q) ( + |B||z|)|z| D f (z) ≤ ϕ(z) + – |ϕ(z)| · | – |z| [(p – q)j–l – ( (A–B)|γ + (p – q)j–l |B|)|z|] –α × Dl+ g (q) (z),
(.)
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which, upon setting |z| = r
and
ϕ(z) = ρ
( ≤ ρ ≤ ),
leads us to the inequality j+ (q) D f (z)
l+ (q) ψ(ρ) D g (z), ≤ (A–B)|γ | ( – r )[(p – q)j–l – ( –α + (p – q)j–l |B|)r] where
(A – B)|γ | ψ(ρ) = –r + |B|r ρ + – r (p – q)j–l – + (p – q)j–l |B| r ρ –α + r + |B|r
(.)
takes its maximum value at ρ = with r = r (p, q, α, γ , j, l, A, B), where r = r (p, q, α, γ , j, l, A, B) is the smallest positive root of equation (.). Furthermore, if ≤ δ ≤ r (p, q, α, γ , j, l, A, B), then the function ψ(ρ) defined by
(A – B)|γ | ψ(ρ) = –δ + |B|δ ρ + – δ (p – q)j–l – + (p – q)j–l |B| δ ρ –α + δ + |B|δ
(.)
is an increasing function on the interval ≤ ρ ≤ so that
(A – B)|γ | j–l j–l + (p – q) |B| δ ψ(ρ) ≤ ψ() = – δ (p – q) – –α ≤ ρ ≤ ; ≤ δ ≤ r (p, q, α, γ , j, l, A, B) .
(.)
Hence, upon setting ρ = in (.), we conclude that (.) of Theorem . holds true for |z| ≤ r (p, q, α, γ , j, l, A, B), which completes the proof of Theorem .. Setting α = in Theorem ., we get the following result. j,l
Corollary . Let the function f ∈ Ap and suppose that g ∈ Sp,q [A, B; γ ]. If Dj f (q) (z) is majorized by Dl g (q) (z) in , and
(p – q)j–l ≥ (A – B)|γ | + (p – q)j–l |B| δ, then j+ (q) l+ (q) D f (z) ≤ D g (z)
|z| ≤ r ,
(.)
Li et al. Journal of Inequalities and Applications 2013, 2013:86 http://www.journalofinequalitiesandapplications.com/content/2013/1/86
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where r = r (p, q, γ , j, l, A, B) is the smallest positive root of the equation
(A – B)|γ | + (p – q)j–l |B| r – (p – q)j–l + |B| r – (A – B)|γ | + (p – q)j–l |B| + r + (p – q)j–l = – ≤ B < A ≤ ; p, j ∈ N; q, l ∈ N ; γ ∈ C ∗ , ≤ δ ≤ r .
(.)
Remark . Corollary . improves the result of Goswami and Aouf [, Theorem ]. Putting p = , q = , j = m, l = n, m > n and γ = in Theorem ., we obtain the following result. Corollary . Let the function f ∈ A and suppose that g ∈ Um,n (α, A, B). If Dm f (z) is majorized by Dn g(z) in , then m+ D f (z) ≤ Dn+ g(z)
|z| ≤ r ,
(.)
where r = r (α, A, B) is the smallest positive root of the equation
A–B A–B + |B| r – + |B| r – + |B| + r + = –α –α (– ≤ B < A ≤ ; ≤ α < ).
(.)
For A = – β, B = –, putting m = , n = and m = , n = in Corollary ., respectively, we obtain the following Corollaries . and .. Corollary . Let the function f ∈ A and suppose that g ∈ US(α, β). If Df (z) is majorized by g(z) in , then
f (z) + zf
(z) ≤ g (z) |z| ≤ r , where r = r (α, β) is the smallest positive root of the equation
( – β) ( – β) + r – r – + r + = ( ≤ α < ; ≤ β < ). –α –α
Corollary . Let the function f ∈ A and suppose that g ∈ UK(α, β). If D f (z) is majorized by Dg(z) in , then D f (z) ≤ D g(z)
|z| ≤ r ,
where r = r (α, β) is the smallest positive root of the equation
( – β) ( – β) + r – r – + r + = ( ≤ α < ; ≤ β < ). –α –α
Also, putting A = , B = –, q = , j = n + and l = n in Theorem ., we obtain the following result.
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Corollary . Let the function f ∈ Ap and suppose that g ∈ Sn (p, α, γ ). If Dn+ f (z) is majorized by Dn g(z) in , then n+ D f (z) ≤ Dn+ g(z)
|z| ≤ r ,
(.)
where r = r (p, α, γ ) is the smallest positive root of the equation
|γ | |γ | + p r – (p + )r – +p+ r+p= –α –α ∗ p ∈ N; γ ∈ C ; ≤ α < .
(.)
Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors jointly worked on the results and they read and approved the final manuscript. Author details 1 School of Mathematics and Statistics, Chifeng University, Chifeng, Inner Mongolia 024000, China. 2 School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China. Acknowledgements Dedicated to Professor Hari M. Srivastava. The present investigation is partly supported by the Natural Science Foundation of Inner Mongolia of People’s Republic of China under Grant 2009MS0113, 2010MS0117. The authors would like to thank the referees for their helpful comments and suggestions to improve our manuscript. Received: 9 November 2012 Accepted: 14 February 2013 Published: 4 March 2013 References 1. MacGregor, TH: Majorization by univalent functions. Duke Math. J. 34, 95-102 (1967) 2. Frasin, BA: Neighborhoods of certain multivalent functions with negative coefficients. Appl. Math. Comput. 193, 1-6 (2007) 3. Salagean, GS: Subclasses of univalent functions. In: Complex Analysis - Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981). Lecture Notes in Math., vol. 1013, pp. 362-372. Springer, Berlin (1983) 4. Goswami, P, Aouf, MK: Majorization properties for certain classes of analytic functions using the Salagean operator. Appl. Math. Lett. 23, 1351-1354 (2010) 5. Li, S-H, Tang, H: Certain new classes of analytic functions defined by using the Salagean operator. Bull. Math. Anal. Appl. 4(2), 62-75 (2010) 6. Kanas, S, Srivastava, HM: Linear operators associated with k-uniformly convex functions. Integral Transforms Spec. Funct. 9, 121-132 (2000) 7. Kanas, S, Yaguchi, T: Subclasses of k-uniformly convex and starlike functions defined by generalized derivative, I. Indian J. Pure Appl. Math. 32(9), 1275-1282 (2001) 8. Kanas, S: Integral operators in classes k-uniformly convex and k-starlike functions. Mathematica (Cluj-Napoca, 1992) 43(66)(1), 77-87 (2001) 9. Kanas, S, Wi´sniowska, A: Conic regions and k-uniform convexity, II. Zeszyty Nauk. Politech. Rzeszowskiej Mat. 170, 65-78 (1998) 10. Kanas, S: Differential subordination related to conic sections. J. Math. Anal. Appl. 317(2), 650-658 (2006) 11. Ramachandran, C, Srivastava, HM, Swaminathan, A: A unified class of k-uniformly convex functions defined by the Salagean derivative operator. Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 55, 47-59 (2007) 12. Shams, S, Kulkarni, SR, Jahangiri, JM: On a class of univalent functions defined by Ruscheweyh derivatives. Kyungpook Math. J. 43, 579-585 (2003) 13. Akbulut, S, Kadioglu, E, Ozdemir, M: On the subclass of p-valently functions. Appl. Math. Comput. 147(1), 89-96 (2004) 14. Nasr, MA, Aouf, MK: Starlike function of complex order. J. Nat. Sci. Math. 25(1), 1-12 (1985) 15. Wiatrowski, P: On the coefficients of some family of holomorphic functions. Zeszyry Nauk. Univ. Lodz. Nauk. Mat.-Przyrod. Ser. II 39, 75-85 (1970) 16. Kanas, S, Darwish, HE: Fekete-Szego problem for starlike and convex functions of complex order. Appl. Math. Lett. 23, 777-782 (2010) 17. Kanas, S, Sugawa, T: On conformal representation of the interior of an ellipse. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 31, 329-348 (2006) 18. Altintas, O, Ozkan, O, Srivastava, HM: Majorization by starlike functions of complex order. Complex Var. Theory Appl. 46, 207-218 (2001) 19. Goyal, SP, Goswami, P: Majorization for certain classes of analytic functions defined by fractional derivatives. Appl. Math. Lett. 22(12), 1855-1858 (2009)
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20. Prajapat, JK, Aouf, MK: Majorization problem for certain class of p-valently analytic function defined by generalized fractional differential operator. Comput. Math. Appl. 63, 42-47 (2012) 21. Nehari, Z: Conformal Mapping. McGraw-Hill, New York (1955)
doi:10.1186/1029-242X-2013-86 Cite this article as: Li et al.: Majorization properties for certain new classes of analytic functions using the Salagean operator. Journal of Inequalities and Applications 2013 2013:86.
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Open Access
Majorization properties for certain new classes of analytic functions using the Salagean operator Shu-Hai Li1* , Huo Tang2,1 and En Ao1 *
Correspondence: [email protected] 1 School of Mathematics and Statistics, Chifeng University, Chifeng, Inner Mongolia 024000, China Full list of author information is available at the end of the article
Abstract In the present paper, we investigate the majorization properties for certain classes of multivalent analytic functions defined by the Salagean operator. Moreover, we point out some new and interesting consequences of our main result. MSC: 30C45 Keywords: analytic functions; multivalent functions; α -uniformly starlike functions of order β ; α -uniformly convex functions of order β ; subordination; majorization property
1 Introduction and definitions Let f and g be two analytic functions in the open unit disk = z ∈ C : |z| < .
(.)
We say that f is majorized by g in (see []) and write f (z) g(z)
(z ∈ )
(.)
if there exists a function ϕ, analytic in , such that ϕ(z) ≤ and
f (z) = ϕ(z)g(z)
(z ∈ ).
(.)
It may be noted here that (.) is closely related to the concept of quasi-subordination between analytic functions. For two functions f and g, analytic in , we say that the function f is subordinate to g in if there exists a Schwarz function ω, which is analytic in with ω() = and
ω(z) <
(z ∈ ),
such that f (z) = g ω(z)
(z ∈ ).
© 2013 Li et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Li et al. Journal of Inequalities and Applications 2013, 2013:86 http://www.journalofinequalitiesandapplications.com/content/2013/1/86
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We denote this subordination by f (z) ≺ g(z). Furthermore, if the function g is univalent in , then f (z) ≺ g(z)
(z ∈ )
⇔
f () = g() and
f () ⊂ g().
Let Ap denote the class of functions of the form ∞
f (z) = zp +
ak z k
p ∈ N = {, , . . .} ,
(.)
k=p+
that are analytic and p-valent in the open unit disk . Also, let A = A. For a function f ∈ Ap , let f (q) denote a qth-order ordinary differential operator by
f
(q)
∞ p! k! p–q z + ak zk–q , (z) = (p – q)! (k – q)!
(.)
k=p+
where p > q, p ∈ N , q ∈ N = N ∪ {} and z ∈ . Next, Frasin [] introduced the differential operator Dm f (q) as follows: Dm f (q) (z) =
∞ p!(p – q)m p–q k!(k – q)m z + ak zk–q . (p – q)! (k – q)!
(.)
k=p+
In view of (.), it is clear that D f () (z) = f (z), D f () (z) = zf (z) and Dm f () (z) = Dm f (z) is a known operator introduced by Salagean []. j,l
Definition . A function f (z) ∈ Ap is said to be in the class Lp,q [A, B; α, γ ] of p-valent functions of complex order γ = in if and only if
j (q)
D f (z) + Az Dj f (q) (z) j–l j–l – α ≺ – (p – q) – (p – q) l (q) l (q) D f (z) γ D f (z) + Bz ∗ z ∈ ; – ≤ B < A ≤ ; j > l; p, j ∈ N; l, q ∈ N ; ≤ α; γ ∈ C = C \ {} .
+
γ
(.)
Clearly, we have the following relationships: j,l j,l () Lp,q [A, B; , γ ] = Sp,q [A, B; γ ]; m,n () L, [A, B; α, ] = Um,n (α, A, B); () L, , [ – β, –; α, ] = US(α, β) ( ≤ β < ) (α-uniformly starlike functions of order β); () L, , [ – β, –; α, ] = UK(α, β) ( ≤ β < ) (α-uniformly convex functions of order β); () Ln+,n p, [, –; α, γ ] = Sn (p, α, γ ) (n ∈ N ); ∗ () L, , [, –; α, γ ] = S(α, γ ) ( ≤ α < , γ ∈ C ); , () L, [, –; α, γ ] = K(α, γ ) ( ≤ α < , γ ∈ C ∗ ); ∗ () L, , [, –; α, – β] = S (α, β) ( ≤ α < , ≤ β < ). j,l
The classes Sp,q [A, B; γ ] and Um,n (α, A, B) were introduced by Goswami and Aouf [] and Li and Tang [], respectively. The classes US(α, β) and UK(α, β) were studied recently
Li et al. Journal of Inequalities and Applications 2013, 2013:86 http://www.journalofinequalitiesandapplications.com/content/2013/1/86
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in [] (see also [–]). The class Sn (p, , γ ) = Sn (p, γ ) was introduced by Akbulut et al. []. Also, the classes S(, γ ) = S(γ ) and K(, γ ) = K(γ ) are said to be classes of starlike and convex of complex order γ = in which were considered by Nasr and Aouf [] and Wiatrowski [] (see also [, ]), and S∗ (, β) = S∗ (β) denotes the class of starlike functions of order β in . A majorization problem for the class S(γ ) has recently been investigated by Altintas j,l et al. []. Also, majorization problems for the classes S∗ (β) and Sp,q [A, B; γ ] have been investigated by MacGregor [] and Goswami and Aouf [], respectively. Very recently, Goyal and Goswami [] (see also []) generalized these results for the fractional derivative operator. In the present paper, we investigate a majorization problem for the class j,l Lp,q [A, B; α, γ ]. j,l
2 Majorization problem for the class Lp,q [A, B; α , γ ] We begin by proving the following result. j,l
Theorem . Let the function f ∈ Ap and suppose that g ∈ Lp,q [A, B; α, γ ]. If Dj f (q) (z) is majorized by Dl g (q) (z) in , and (p – q)j–l ≥
(A – B)|γ | + (p – q)j–l |B| δ, –α
then j+ (q) l+ (q) D f (z) ≤ D g (z)
|z| ≤ r ,
(.)
where r = r (p, q, α, γ , j, l, A, B) is the smallest positive root of the equation
(A – B)|γ | + (p – q)j–l |B| r – (p – q)j–l + |B| r –α (A – B)|γ | + (p – q)j–l |B| + r + (p – q)j–l = – –α – ≤ B < A ≤ ; p, j ∈ N; q, l ∈ N ; ≤ α < ; γ ∈ C ∗ , ≤ δ ≤ r . j,l
Proof Suppose that g ∈ Lp,q [A, B; α, γ ]. Then, making use of the fact that – α| – | ≺
+ Az + Bz
⇔
+ Az – αe–iφ + αe–iφ ≺ + Bz
and letting =+ γ
Dj g (q) (z) – (p – q)j–l Dl g (q) (z)
in (.), we obtain +
γ
Dj g (q) (z) – (p – q)j–l Dl g (q) (z)
+ Az – αe–iφ + αe–iφ ≺ + Bz
(φ ∈ R),
(.)
Li et al. Journal of Inequalities and Applications 2013, 2013:86 http://www.journalofinequalitiesandapplications.com/content/2013/1/86
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or, equivalently, + γ
–iφ
+ ( A–αBe )z Dj g (q) (z) –αe–iφ j–l ≺ – (p – q) l (q) D g (z) + Bz
(.)
which holds true for all z ∈ . We find from (.) that + γ
–iφ
+ ( A–αBe )ω(z) Dj g (q) (z) –αe–iφ j–l = – (p – q) , Dl g (q) (z) + Bω(z)
(.)
where ω(z) = c z+c z +· · · , ω ∈ P, P denotes the well-known class of the bounded analytic functions in and satisfies the conditions ω() = and
ω(z) ≤ |z|
(z ∈ ).
From (.), we get (A–B)γ j–l j–l Dj g (q) (z) (p – q) + [ –αe–iφ + (p – q) B]ω(z) = . Dl g (q) (z) + Bω(z)
(.)
By virtue of (.), we obtain l (q) D g (z) ≤ ≤
+ |B||z| (p – q)j–l
(A–B)γ – | –αe –iφ
+ (p – q)j–l B||z|
j (q) D g (z)
+ |B||z| | (p – q)j–l – [ (A–B)|γ –α
+ (p – q)j–l |B|]|z|
j (q) D g (z).
(.)
Next, since Dj f (q) (z) is majorized by Dl g (q) (z) in , thus from (.), we have Dj f (q) (z) = ϕ(z)Dl g (q) (z). Differentiating the above equality with respect to z and multiplying by z, we get Dj+ f (q) (z) = zϕ (z)Dl g (q) (z) + ϕ(z)Dl+ g (q) (z).
(.)
Thus, by noting that ϕ(z) ∈ P satisfies the inequality (see, e.g., Nehari []) – |ϕ(z)| ϕ (z) ≤ – |z|
(z ∈ )
(.)
and making use of (.) and (.) in (.), we obtain
j+ (q) ( + |B||z|)|z| D f (z) ≤ ϕ(z) + – |ϕ(z)| · | – |z| [(p – q)j–l – ( (A–B)|γ + (p – q)j–l |B|)|z|] –α × Dl+ g (q) (z),
(.)
Li et al. Journal of Inequalities and Applications 2013, 2013:86 http://www.journalofinequalitiesandapplications.com/content/2013/1/86
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which, upon setting |z| = r
and
ϕ(z) = ρ
( ≤ ρ ≤ ),
leads us to the inequality j+ (q) D f (z)
l+ (q) ψ(ρ) D g (z), ≤ (A–B)|γ | ( – r )[(p – q)j–l – ( –α + (p – q)j–l |B|)r] where
(A – B)|γ | ψ(ρ) = –r + |B|r ρ + – r (p – q)j–l – + (p – q)j–l |B| r ρ –α + r + |B|r
(.)
takes its maximum value at ρ = with r = r (p, q, α, γ , j, l, A, B), where r = r (p, q, α, γ , j, l, A, B) is the smallest positive root of equation (.). Furthermore, if ≤ δ ≤ r (p, q, α, γ , j, l, A, B), then the function ψ(ρ) defined by
(A – B)|γ | ψ(ρ) = –δ + |B|δ ρ + – δ (p – q)j–l – + (p – q)j–l |B| δ ρ –α + δ + |B|δ
(.)
is an increasing function on the interval ≤ ρ ≤ so that
(A – B)|γ | j–l j–l + (p – q) |B| δ ψ(ρ) ≤ ψ() = – δ (p – q) – –α ≤ ρ ≤ ; ≤ δ ≤ r (p, q, α, γ , j, l, A, B) .
(.)
Hence, upon setting ρ = in (.), we conclude that (.) of Theorem . holds true for |z| ≤ r (p, q, α, γ , j, l, A, B), which completes the proof of Theorem .. Setting α = in Theorem ., we get the following result. j,l
Corollary . Let the function f ∈ Ap and suppose that g ∈ Sp,q [A, B; γ ]. If Dj f (q) (z) is majorized by Dl g (q) (z) in , and
(p – q)j–l ≥ (A – B)|γ | + (p – q)j–l |B| δ, then j+ (q) l+ (q) D f (z) ≤ D g (z)
|z| ≤ r ,
(.)
Li et al. Journal of Inequalities and Applications 2013, 2013:86 http://www.journalofinequalitiesandapplications.com/content/2013/1/86
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where r = r (p, q, γ , j, l, A, B) is the smallest positive root of the equation
(A – B)|γ | + (p – q)j–l |B| r – (p – q)j–l + |B| r – (A – B)|γ | + (p – q)j–l |B| + r + (p – q)j–l = – ≤ B < A ≤ ; p, j ∈ N; q, l ∈ N ; γ ∈ C ∗ , ≤ δ ≤ r .
(.)
Remark . Corollary . improves the result of Goswami and Aouf [, Theorem ]. Putting p = , q = , j = m, l = n, m > n and γ = in Theorem ., we obtain the following result. Corollary . Let the function f ∈ A and suppose that g ∈ Um,n (α, A, B). If Dm f (z) is majorized by Dn g(z) in , then m+ D f (z) ≤ Dn+ g(z)
|z| ≤ r ,
(.)
where r = r (α, A, B) is the smallest positive root of the equation
A–B A–B + |B| r – + |B| r – + |B| + r + = –α –α (– ≤ B < A ≤ ; ≤ α < ).
(.)
For A = – β, B = –, putting m = , n = and m = , n = in Corollary ., respectively, we obtain the following Corollaries . and .. Corollary . Let the function f ∈ A and suppose that g ∈ US(α, β). If Df (z) is majorized by g(z) in , then
f (z) + zf
(z) ≤ g (z) |z| ≤ r , where r = r (α, β) is the smallest positive root of the equation
( – β) ( – β) + r – r – + r + = ( ≤ α < ; ≤ β < ). –α –α
Corollary . Let the function f ∈ A and suppose that g ∈ UK(α, β). If D f (z) is majorized by Dg(z) in , then D f (z) ≤ D g(z)
|z| ≤ r ,
where r = r (α, β) is the smallest positive root of the equation
( – β) ( – β) + r – r – + r + = ( ≤ α < ; ≤ β < ). –α –α
Also, putting A = , B = –, q = , j = n + and l = n in Theorem ., we obtain the following result.
Li et al. Journal of Inequalities and Applications 2013, 2013:86 http://www.journalofinequalitiesandapplications.com/content/2013/1/86
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Corollary . Let the function f ∈ Ap and suppose that g ∈ Sn (p, α, γ ). If Dn+ f (z) is majorized by Dn g(z) in , then n+ D f (z) ≤ Dn+ g(z)
|z| ≤ r ,
(.)
where r = r (p, α, γ ) is the smallest positive root of the equation
|γ | |γ | + p r – (p + )r – +p+ r+p= –α –α ∗ p ∈ N; γ ∈ C ; ≤ α < .
(.)
Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors jointly worked on the results and they read and approved the final manuscript. Author details 1 School of Mathematics and Statistics, Chifeng University, Chifeng, Inner Mongolia 024000, China. 2 School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China. Acknowledgements Dedicated to Professor Hari M. Srivastava. The present investigation is partly supported by the Natural Science Foundation of Inner Mongolia of People’s Republic of China under Grant 2009MS0113, 2010MS0117. The authors would like to thank the referees for their helpful comments and suggestions to improve our manuscript. Received: 9 November 2012 Accepted: 14 February 2013 Published: 4 March 2013 References 1. MacGregor, TH: Majorization by univalent functions. Duke Math. J. 34, 95-102 (1967) 2. Frasin, BA: Neighborhoods of certain multivalent functions with negative coefficients. Appl. Math. Comput. 193, 1-6 (2007) 3. Salagean, GS: Subclasses of univalent functions. In: Complex Analysis - Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981). Lecture Notes in Math., vol. 1013, pp. 362-372. Springer, Berlin (1983) 4. Goswami, P, Aouf, MK: Majorization properties for certain classes of analytic functions using the Salagean operator. Appl. Math. Lett. 23, 1351-1354 (2010) 5. Li, S-H, Tang, H: Certain new classes of analytic functions defined by using the Salagean operator. Bull. Math. Anal. Appl. 4(2), 62-75 (2010) 6. Kanas, S, Srivastava, HM: Linear operators associated with k-uniformly convex functions. Integral Transforms Spec. Funct. 9, 121-132 (2000) 7. Kanas, S, Yaguchi, T: Subclasses of k-uniformly convex and starlike functions defined by generalized derivative, I. Indian J. Pure Appl. Math. 32(9), 1275-1282 (2001) 8. Kanas, S: Integral operators in classes k-uniformly convex and k-starlike functions. Mathematica (Cluj-Napoca, 1992) 43(66)(1), 77-87 (2001) 9. Kanas, S, Wi´sniowska, A: Conic regions and k-uniform convexity, II. Zeszyty Nauk. Politech. Rzeszowskiej Mat. 170, 65-78 (1998) 10. Kanas, S: Differential subordination related to conic sections. J. Math. Anal. Appl. 317(2), 650-658 (2006) 11. Ramachandran, C, Srivastava, HM, Swaminathan, A: A unified class of k-uniformly convex functions defined by the Salagean derivative operator. Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 55, 47-59 (2007) 12. Shams, S, Kulkarni, SR, Jahangiri, JM: On a class of univalent functions defined by Ruscheweyh derivatives. Kyungpook Math. J. 43, 579-585 (2003) 13. Akbulut, S, Kadioglu, E, Ozdemir, M: On the subclass of p-valently functions. Appl. Math. Comput. 147(1), 89-96 (2004) 14. Nasr, MA, Aouf, MK: Starlike function of complex order. J. Nat. Sci. Math. 25(1), 1-12 (1985) 15. Wiatrowski, P: On the coefficients of some family of holomorphic functions. Zeszyry Nauk. Univ. Lodz. Nauk. Mat.-Przyrod. Ser. II 39, 75-85 (1970) 16. Kanas, S, Darwish, HE: Fekete-Szego problem for starlike and convex functions of complex order. Appl. Math. Lett. 23, 777-782 (2010) 17. Kanas, S, Sugawa, T: On conformal representation of the interior of an ellipse. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 31, 329-348 (2006) 18. Altintas, O, Ozkan, O, Srivastava, HM: Majorization by starlike functions of complex order. Complex Var. Theory Appl. 46, 207-218 (2001) 19. Goyal, SP, Goswami, P: Majorization for certain classes of analytic functions defined by fractional derivatives. Appl. Math. Lett. 22(12), 1855-1858 (2009)
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doi:10.1186/1029-242X-2013-86 Cite this article as: Li et al.: Majorization properties for certain new classes of analytic functions using the Salagean operator. Journal of Inequalities and Applications 2013 2013:86.
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