Open Access

Some subordination and superordination results of generalized Srivastava-Attiya operator

  • Mohamed Kamal Aouf1,
  • Adela Osman Mostafa1,
  • Awatif Mohamed Shahin1 and
  • Samar Mohamed Madian1Email author
Journal of Inequalities and Applications20122012:115

https://doi.org/10.1186/1029-242X-2012-115

Received: 8 December 2011

Accepted: 24 May 2012

Published: 24 May 2012

Abstract

In this article, we obtain some subordination and superordination-preserving results of the generalized Srivastava-Attyia operator. Sandwich-type result is also obtained.

Mathematics Subject Classification 2000: 30C45.

Keywords

analytic functionHadamard productdifferential subordinationsuperordination

1 Introduction

Let H(U) be the class of functions analytic in U = { z : | z | < 1 } and H[a, n] be the subclass of H(U) consisting of functions of the form f(z) = a + a n z n + a n +1zn+1 + ..., with H0 = H[0, 1] and H = H[1, 1]. Denote A(p) by the class of all analytic functions of the form
f ( z ) = z p + n = 1 a p + n z p + n ( p = { 1 , 2 , 3 , } ; z U )
(1.1)

and let A (1) = A. For f, F H(U), the function f(z) is said to be subordinate to F(z), or F(z) is superordinate to f(z), if there exists a function ω(z) analytic in U with ω(0) = 0 and (z)| < 1(z U), such that f(z) = F(ω(z)). In such a case we write f(z) F(z). If F is univalent, then f(z) F(z) if and only if f(0) = F(0) and f(U) F(U) (see [1, 2]).

Let ϕ : 2 × U and h(z) be univalent in U. If p(z) is analytic in U and satisfies the first order differential subordination:
ϕ p ( z ) , z p ( z ) ; z h ( z ) ,
(1.2)
then p (z) is a solution of the differential subordination (1.2). The univalent function q (z) is called a dominant of the solutions of the differential subordination (1.2) if p(z) q(z) for all p(z) satisfying (1.2). A univalent dominant q ̃ that satisfies q ̃ q for all dominants of (1.2) is called the best dominant. If p(z) and ϕ(p(z), zp' (z) ; z) are univalent in U and if p(z) satisfies the first order differential superordination:
h ( z ) ϕ p ( z ) , z p ( z ) ; z ,
(1.3)

then p(z) is a solution of the differential superordination (1.3). An analytic function q(z) is called a subordinant of the solutions of the differential superordination (1.3) if q(z) p(z) for all p(z) satisfying (1.3). A univalent subordinant q ̃ that satisfies q q ̃ for all subordinants of (1.3) is called the best subordinant (see [1, 2]).

The general Hurwitz-Lerch Zeta function Φ(z, s, a) is defined by:
Φ ( z , s , a ) = n = 0 z n ( n + a ) s ,
(1.4)

( a \ 0 - ; 0 - = { 0 , - 1 , - 2 , } ; s when |z| < 1; R{s} > 1 when |z| = 1).

For further interesting properties and characteristics of the Hurwitz-Lerch Zeta function Φ(z, s, a) (see [37]).

Recently, Srivastava and Attiya [8] introduced the linear operator L s,b : AA, defined in terms of the Hadamard product by
L s , b ( f ) ( z ) = G s , b ( z ) * f ( z ) ( z U ; b \ 0 - ; s ) ,
(1.5)
where
G s , b = ( 1 + b ) s [ Φ ( z , s , b ) - b - s ] ( z U ) .
(1.6)

The Srivastava-Attiya operator L s,b contains among its special cases, the integral operators introduced and investigated by Alexander [9], Libera [10] and Jung et al. [11].

Analogous to L s,b , Liu [12] defined the operator J p,s,b : A(p) → A(p) by
J p , s , b ( f ) ( z ) = G p , s , b ( z ) * f ( z ) ( z U ; b 0 - ; s ; p ) ,
(1.7)
where
G p , s , b = ( 1 + b ) s [ Φ p ( z , s , b ) - b - s ]
and
Φ p ( z , s , b ) = 1 b s + n = 0 z n + p ( n + 1 + b ) s .
(1.8)
It is easy to observe from (1.7) and (1.8) that
J p , s , b ( f ) ( z ) = z p + n = 1 1 + b n + 1 + b s a n + p z n + p .
(1.9)
We note that
  1. (i)

    J p ,0, b (f)(z) = f (z);

     
  2. (ii)

    J 1 , s , b ( f ) ( z ) = L s , b f ( z ) ( s , b \ 0 - ) , where the operator L s,b was introduced by Srivastava and Attiya [8];

     
  3. (iii)

    J p , 1 , v + p - 1 f z = F v , p f z ( v > - p , p ) , where the operator F v,p was introduced by Choi et al. [13];

     
  4. (iv)

    J p , α , p ( f ) ( z ) = I p α f ( z ) ( α 0 , p ) , where the operator I p α was introduced by Shams et al. [14];

     
  5. (v)

    J p , m , p - 1 ( f ) ( z ) = J p m f ( z ) ( m 0 = { 0 } , p ) , where the operator J p m was introduced by El-Ashwah and Aouf [15];

     
  6. (vi)

    J p , m , p + l - 1 ( f ) ( z ) = J p m ( l ) f ( z ) ( m 0 , p , l 0 ) , where the operator J p m ( l ) was introduced by El-Ashwah and Aouf [15].

     
It follows from (1.9) that:
z ( J p , s  + 1 , b ( f )( z ))' = ( b  + 1) J p , s , b ( f )( z ) { ( b  + 1 {  p ) J p , s  + 1 , b ( f )( z ) .
(1.10)

To prove our results, we need the following definitions and lemmas.

Definition 1[1]Denote by the set of all functions q(z) that are analytic and injective on Ū \ E ( q ) where
E ( q ) = ζ U : lim z ζ q ( z ) =

and are such that q'(ζ) 0 for ζ δU\E(q). Further let the subclass of for which q(0) = a be denoted by F a , F 0 F 0 and F 1 F 1 .

Definition 2[2]A function L (z, t) (z U, t ≥ 0) is said to be a subordination chain if L (0, t) is analytic and univalent in U for all t ≥ 0, L (z, 0) is continuously differentiable on [0; 1) for all z U and L (z, t1) L (z, t2) for all 0 ≤ t1t2.

Lemma 1[16]The function L z , t : U × 0 ; 1 of the form
L ( z , t ) = a 1 ( t ) z + a 2 ( t ) z 2 + ( a 1 ( t ) 0 ; t 0 )
and lim t | a 1 ( t ) | = is a subordination chain if and only if
Re z L ( z , t ) / z L ( z , t ) / t > 0 ( z U , t 0 ) .
Lemma 2[17]Suppose that the function H : 2 satisfies the condition
Re { H ( i s ; t ) } 0
for all real s and for all t-n (1 + s2) / 2, n . If the function p(z) = 1+p n z n +p n +1zn+1+ ...is analytic in U and
Re H p ( z ) ; z p ( z ) > 0 ( z U ) ,

then Re {p(z)} > 0 for z U.

Lemma 3[18]Let κ, γ with κ ≠ 0 and let h H(U) with h(0) = c. If Re {κh(z) + γ} > 0 (z U), then the solution of the following differential equation:
q ( z ) + z q ( z ) κ q ( z ) + γ = h ( z ) ( z U ; q ( 0 ) = c )

is analytic in U and satisfies Re {κq(z) + γ} > 0 for z U.

Lemma 4[1]Let p F a and let q(z) = a + a n z n + an+1zn+1 + ...be analytic in U with q (z) ≠ a and n ≥ 1. If q is not subordinate to p, then there exists two points z0 = r0e U and ζ0 δU\E(q) such that
q ( U r 0 ) p ( U ) ; q ( z 0 ) = p ( ζ 0 ) a n d z 0 p ( z 0 ) = m ζ 0 p ( ζ 0 ) ( m n ) .
Lemma 5[2]Let q H[a; 1] and ϕ : 2 . Also set φ(q(z), zq'(z)) = h(z). If L(z, t) = φ (q (z), tzq'(z)) is a subordination chain and q H a ; 1 F a , then
h ( z ) φ q ( z ) , z q ( z ) ,

implies that q(z) p(z). Furthermore, if φ(q(z), zq'(z)) = h(z) has a univalent solution q F a , then q is the best subordinant.

In the present article, we aim to prove some subordination-preserving and superordination-preserving properties associated with the integral operator J p,s,b . Sandwich-type result involving this operator is also derived.

2 Main results

Unless otherwise mentioned, we assume throughout this section that b \ 0 - , s , Re {b}, µ > 0, p , z U and the powers are understood as principle values.

Theorem 1. Let f, g A (p) and
Re 1 + z ϕ ( z ) ϕ ( z ) > - δ ϕ ( z ) = J p , s - 1 , b ( g ) ( z ) J p , s , b ( g ) ( z ) J p , s , b ( g ) ( z ) z p μ ; z U ,
(2.1)
where δ is given by
δ = 1 + μ 2 | b + 1 | 2 - | 1 - μ 2 ( b + 1 ) 2 | 4 μ [ 1 + Re { b } ] ( z U ) .
(2.2)
Then the subordination condition
J p , s - 1 , b ( f ) ( z ) J p , s , b ( f ) ( z ) J p , s , b ( f ) ( z ) z p μ J p , s - 1 , b ( g ) ( z ) J p , s , b ( g ) ( z ) J p , s , b ( g ) ( z ) z p μ ,
(2.3)
implies that
J p , s , b ( f ) ( z ) z p μ J p , s , b ( g ) ( z ) z p μ ,
(2.4)

where J p , s , b ( g ) ( z ) z p μ is the best dominant.

Proof. Let us define the functions F(z) and G(z) in U by
F ( z ) = J p , s , b ( f ) ( z ) z p μ and G ( z ) = J p , s , b ( g ) ( z ) z p μ ( z U )
(2.5)
and without loss of generality we assume that G(z) is analytic, univalent on and
G ( ζ ) 0 ( | ζ | = 1 ) .

If not, then we replace F(z) and G(z) by F(ρz) and G(ρz), respectively, with 0 < ρ < 1. These new functions have the desired properties on , so we can use them in the proof of our result and the results would follow by letting ρ → 1.

We first show that, if
q ( z ) = 1 + z G ( z ) G ( z ) ( z U ) ,
(2.6)
then
Re { q ( z ) } > 0 ( z U ) .
From (1.10) and the definition of the functions G, ϕ, we obtain that
ϕ ( z ) = G ( z ) + z G ( z ) μ ( b + 1 ) .
(2.7)
Differentiating both sides of (2.7) with respect to z yields
ϕ ( z ) = 1 + 1 μ ( b + 1 ) G ( z ) + z G ( z ) μ ( b + 1 ) .
(2.8)
Combining (2.6) and (2.8), we easily get
1 + z ϕ ( z ) ϕ ( z ) = q ( z ) + z q ( z ) q ( z ) + μ ( b + 1 ) = h ( z ) ( z U ) .
(2.9)
It follows from (2.1) and (2.9) that
Re { h ( z ) + μ ( b + 1 ) } > 0 ( z U ) .
(2.10)
Moreover, by using Lemma 3, we conclude that the differential Equation (2.9) has a solution q(z) H(U) with h(0) = q(0) = 1. Let
H ( u , v ) = u + v u + μ ( b + 1 ) + δ ,

Where δ is given by (2.2). From (2.9) and (2.10), we obtain Re { H ( q ( z ) ; ) z q ( z ) ) } > 0 ( z U ) .

To verify the condition
Re { H ( i ϑ ; t ) } 0 ϑ ; t - 1 + ϑ 2 2 ,
(2.11)
we proceed as follows:
Re { H ( i ϑ ; t ) } = Re i ϑ + t μ ( b + 1 ) + i ϑ + δ = t μ ( 1 + Re ( b ) ) | μ ( b + 1 ) + i ϑ | 2 + δ - ϒ ( b , ϑ , δ ) 2 | μ ( b + 1 ) + i ϑ | 2 ,
where
ϒ ( b , ϑ , δ ) = [ μ ( 1 + Re ( b ) ) - 2 δ ] ϑ 2 - 4 δ μ Im ( b ) ϑ - 2 δ | μ ( b + 1 ) | 2 + μ ( 1 + Re { b } ) .
(2.12)
For δ given by (2.2), the coefficient of ϑ2 in the quadratic expression ϒ(b, ϑ, δ) given by (2.12) is positive or equal to zero. To check this, put µ(b + 1) = c, so that
μ ( 1 + Re ( b ) ) = c 1 and μ Im ( b ) = c 2 .
We thus have to verify that
c 1 - 2 δ 0 ,
or
c 1 2 δ = 1 + | c | 2 - | 1 - c 2 | 2 c 1 .
This inequality will hold true if
2 c 1 2 + | 1 - c 2 | 1 + | c | 2 = 1 + c 1 2 + c 2 2 ,
that is, if
| 1 - c 2 | 1 - Re ( c 2 ) ,
which is obviously true. Moreover, the quadratic expression ϒ(b, ϑ, δ)by ϑ in (2.12) is a perfect square for the assumed value of δ given by (2.2). Hence we see that (2.11) holds. Thus, by Lemma 2, we conclude that
Re { q ( z ) } > 0 ( z U ) ,
that is, that G defined by (2.5) is convex (univalent) in U. Next, we prove that the subordination condition (2.3) implies that
F ( z ) G ( z ) ,
for the functions F and G defined by (2.5). Consider the function L(z, t) given by
L ( z , t ) = G ( z ) + ( 1 + t ) z G ( z ) μ ( b + 1 ) ( 0 t < ; z U ) .
(2.13)
We note that
L ( z , t ) z z = 0 = G ( 0 ) 1 + 1 + t μ ( b + 1 ) 0 ( 0 t < ; z U ; Re { μ ( b + 1 ) } > 0 ) .
This show that the function
L ( z , t ) = a 1 ( t ) z +
satisfies the condition a1 (t) ≠ 0 (0 ≤ t < ∞). Further, we have
Re z L ( z , t ) / z L ( z , t ) / t = Re { μ ( b + 1 ) + ( 1 + t ) q ( z ) } > 0 ( 0 t < ; z U ) .
Since G(z) is convex and Re {µ(b + 1)} > 0. Therefore, by using Lemma 1, we deduce that L(z, t) is a subordination chain. It follows from the definition of subordination chain that
ϕ ( z ) = G ( z ) + z G ( z ) μ ( b + 1 ) = L ( z , 0 )
and
L ( z , 0 ) L ( z , t ) ( 0 t < ) ,
which implies that
L ( ζ , t ) L ( U , 0 ) = ϕ ( U ) ( 0 t < ; ζ U ) .
(2.14)
If F is not subordinate to G, by using Lemma 4, we know that there exist two points z0 U and ζ0 ∂U such that
F ( z 0 ) = G ( ζ 0 ) a n d z 0 F ( z 0 ) = ( 1 + t ) ζ 0 G ( ζ 0 ) ( 0 t < ) .
(2.15)
Hence, by using (2.5), (2.13), (2.15) and (2.3), we have
L ( ζ 0 , t ) = G ( ζ 0 ) + ( 1 + t ) ζ 0 G ( ζ 0 ) μ ( b + 1 ) = F ( z 0 ) + z 0 F ( z 0 ) μ ( b + 1 ) = J p , s - 1 , b ( f ) ( z ) J p , s , b ( f ) ( z ) J p , s , b ( f ) ( z ) z p μ ϕ ( U ) .

This contradicts (2.14). Thus, we deduce that F G. Considering F = G, we see that the function G is the best dominant. This completes the proof of Theorem 1.

We now derive the following superordination result.

Theorem 2. Let f, g A (p) and
Re 1 + z ϕ ( z ) ϕ ( z ) > - δ ϕ ( z ) = J p , s - 1 , b ( g ) ( z ) J p , s , b ( g ) ( z ) J p , s , b ( g ) ( z ) z p μ ; z U ,
(2.16)
where δ is given by (2.2) . If the function J p , s - 1 , b ( f ) ( z ) J p , s , b ( f ) ( z ) J p , s , b ( f ) ( z ) z p μ is univalent in U and J p , s , b ( f ) ( z ) z p μ F , then the superordination condition
J p , s - 1 , b ( g ) ( z ) J p , s , b ( g ) ( z ) J p , s , b ( g ) ( z ) z p μ J p , s - 1 , b ( f ) ( z ) J p , s , b ( f ) ( z ) J p , s , b ( f ) ( z ) z p μ ,
(2.17)
implies that
J p , s , b ( g ) ( z ) z p μ J p , s , b ( f ) ( z ) z p μ ,
(2.18)

where J p , s , b ( f ) ( z ) z p μ is the best subordinant.

Proof. Suppose that the functions F, G and q are defined by (2.5) and (2.6), respectively. By applying similar method as in the proof of Theorem 1, we get
Re { q ( z ) } > 0 ( z U ) .
Next, to arrive at our desired result, we show that G F. For this, we suppose that the function L(z, t) be defined by (2.13). Since G is convex, by applying a similar method as in Theorem 1, we deduce that L(z, t) is subordination chain. Therefore, by using Lemma 5, we conclude that G F. Moreover, since the differential equation
ϕ ( z ) = G ( z ) + z G ( z ) μ ( b + 1 ) = φ G ( z ) , z G ( z )

has a univalent solution G, it is the best subordinant. This completes the proof of Theorem 2.

Combining the above-mentioned subordination and superordination results involving the operator J p,s,b , the following "sandwich-type result" is derived.

Theorem 3. Let f, g j A (p) (j = 1, 2) and
Re 1 + z ϕ j ( z ) ϕ j ( z ) > - δ ϕ j ( z ) = J p , s - 1 , b ( g j ) ( z ) J p , s , b ( g j ) ( z ) J p , s , b ( g j ) ( z ) z p μ ( j = 1 , 2 ) ; z U ,
where δ is given by (2.2) . If the function J p , s - 1 , b ( f ) ( z ) J p , s , b ( f ) ( z ) J p , s , b ( f ) ( z ) z p μ is univalent in U and J p , s , b ( f ) ( z ) z p μ F , then the condition
J p , s - 1 , b ( g 1 ) ( z ) J p , s , b ( g 1 ) ( z ) J p , s , b ( g 1 ) ( z ) z p μ J p , s - 1 , b ( f ) ( z ) J p , s , b ( f ) ( z ) J p , s , b ( f ) ( z ) z p μ J p , s - 1 , b ( g 2 ) ( z ) J p , s , b ( g 2 ) ( z ) J p , s , b ( g 2 ) ( z ) z p μ ,
(2.19)
implies that
J p , s , b ( g 1 ) ( z ) z p μ J p , s , b ( f ) ( z ) z p μ J p , s , b ( g 2 ) ( z ) z p μ ,
(2.20)

where J p , s , b ( g 1 ) ( z ) z p μ and J p , s , b ( g 2 ) ( z ) z p μ are, respectively, the best subordinant and the best dominant.

Remark. (i) Putting µ = 1, b = p and s = α(α = 0, p ) in our results of this article, we obtain the results obtained by Aouf and Seoudy[19];

(ii) Specializing the parameters s and b in our results of this article, we obtain the results for the corresponding operators F v,p , I p α , J p m and J p m ( l ) which are defined in the introduction.

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Mansoura University

References

  1. Miller SS, Mocanu PT: Differential Subordinations: Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics. Volume 225. Marcel Dekker, New York/Basel; 2000.Google Scholar
  2. Miller S, Mocanu PT: Subordinants of differential superordinations. Complex Var Theory Appl 2003, 48(10):815–826. 10.1080/02781070310001599322MathSciNetView ArticleMATHGoogle Scholar
  3. Choi JH, Srivastava HM: Certain families of series associated with the Hurwitz-Lerch Zeta function. Appl Math Comput 2005, 170: 399–409. 10.1016/j.amc.2004.12.004MathSciNetView ArticleMATHGoogle Scholar
  4. Lin S-D, Srivastava HM: Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations. Appl Math Comput 2004, 154: 725–733. 10.1016/S0096-3003(03)00746-XMathSciNetView ArticleMATHGoogle Scholar
  5. Lin S-D, Srivastava HM, Wang P-Y: Some expansion formulas for a class of generalized Hurwitz-Lerch Zeta functions. Integr Trans Spec Funct 2006, 17: 817–827. 10.1080/10652460600926923MathSciNetView ArticleMATHGoogle Scholar
  6. Luo Q-M, Srivastava HM: Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials. J Math Anal Appl 2005, 308: 290–302. 10.1016/j.jmaa.2005.01.020MathSciNetView ArticleMATHGoogle Scholar
  7. Srivastava HM, Choi J: Series Associated with the Zeta and Related Functions. Kluwer Academic Publishers, Dordrecht; 2001.View ArticleMATHGoogle Scholar
  8. Srivastava HM, Attiya AA: An integral operator associated with the Hurwitz-Lerch Zeta function and differential subordination. Integr Trans Spec Funct 2007, 18: 207–216. 10.1080/10652460701208577MathSciNetView ArticleMATHGoogle Scholar
  9. Alexander JW: Functions which map the interior of the unit circle upon simple regions. Ann Math Ser 1915, 17(2):12–22.View ArticleMathSciNetMATHGoogle Scholar
  10. Libera RJ: Some classes of regular univalent functions. Proc Am Math Soc 1969, 16: 755–758.MathSciNetView ArticleMATHGoogle Scholar
  11. Jung TB, Kim YC, Srivastava HM: The Hardy space of analytic functions associated with certain one-parameter families of integral operators. J Math Anal Appl 1993, 176: 138–147. 10.1006/jmaa.1993.1204MathSciNetView ArticleMATHGoogle Scholar
  12. Liu J-L: Subordinations for certain multivalent analytic functions associated with the generalized Srivastava-Attiya operator. Integr Trans Spec Funct 2007, 18: 207–216. 10.1080/10652460701208577View ArticleGoogle Scholar
  13. Choi JH, Saigo M, Srivastava HM: Some inclusion properties of a certain family of integral operators. J Math Anal Appl 2002, 276: 432–445. 10.1016/S0022-247X(02)00500-0MathSciNetView ArticleMATHGoogle Scholar
  14. Shams S, Kulkarni SR, Jahangiri JM: Subordination properties of p -valent functions defined by integral operators. Int J Math Math Sci 2006, 2006: 1. 3 (Article ID 94572)MathSciNetView ArticleMATHGoogle Scholar
  15. El-Ashwah RM, Aouf MK: Some properties of new integral operator. Acta Univ Apulensis 2010, 24: 51–61 (2010).MathSciNetMATHGoogle Scholar
  16. Pommerenke Ch: Univalent Functions. Vandenhoeck and Ruprecht, Göttingen 1975.Google Scholar
  17. Miller SS, Mocanu PT: Differential subordinations and univalent functions. Michigan Math J 1981, 28(2):157–172.MathSciNetView ArticleMATHGoogle Scholar
  18. Miller SS, Mocanu PT: Univalent solutions of Briot-Bouquet differential equations. J Diff Equ 1985, 56(3):297–309. 10.1016/0022-0396(85)90082-8MathSciNetView ArticleMATHGoogle Scholar
  19. Aouf MK, Seoudy TM: Some preserving subordination and superordination results of certain integral operator. Int J Open Probl Complex Analysis 2011, 3(3):1–8.MathSciNetMATHGoogle Scholar

Copyright

© Aouf et al; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. 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.