# Some subordination and superordination results of generalized Srivastava-Attiya operator

## 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.

## 1 Introduction

Let H(U) be the class of functions analytic in $U= { z ∈ ℂ : | z | < 1 }$ and H[a, n] be the subclass of H(U) consisting of functions of the form f(z) = a + a n zn + 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

$f ( z ) = z p + ∑ n = 1 ∞ a p + n z p + n ( p ∈ ℕ = { 1 , 2 , 3 , … } ; z ∈ U )$
(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 $ϕ: ℂ 2 ×U→ℂ$ 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)

$( a ∈ ℂ \ ℤ 0 - ; ℤ 0 - = { 0 , - 1 , - 2 , … }$; $s∈ℂ$ 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 ).

Recently, Srivastava and Attiya  introduced the linear operator L s,b : AA, defined in terms of the Hadamard product by

$L s , b ( f ) ( z ) = G s , b ( z ) * f ( z ) ( z ∈ U ; b ∈ ℂ \ ℤ 0 - ; s ∈ ℂ ) ,$
(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 , Libera  and Jung et al. .

Analogous to L s,b , Liu  defined the operator J p,s,b : A(p) → A(p) by

$J p , s , b ( f ) ( z ) = G p , s , b ( z ) *f ( z ) ( z ∈ U ; b ∈ ℂ ℤ 0 - ; s ∈ ℂ ; p ∈ ℕ ) ,$
(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)

$J 1 , s , b ( f ) ( z ) = L s , b f ( z ) ( s ∈ ℂ , b ∈ ℂ \ ℤ 0 - )$, where the operator L s,b was introduced by Srivastava and Attiya ;

3. (iii)

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

4. (iv)

$J p , α , p ( f ) ( z ) = I p α f ( z ) ( α ≥ 0 , p ∈ ℕ )$, where the operator $I p α$ was introduced by Shams et al. ;

5. (v)

$J p , m , p - 1 ( f ) ( z ) = J p m f ( z ) ( m ∈ ℕ 0 = ℕ ∪ { 0 } , p ∈ ℕ )$, where the operator $J p m$was introduced by El-Ashwah and Aouf ;

6. (vi)

$J p , m , p + l - 1 ( f ) ( z ) = J p m ( l ) f ( z ) ( m ∈ ℕ 0 , p ∈ ℕ , l ≥ 0 )$, where the operator $J p m ( l )$ was introduced by El-Ashwah and Aouf .

It follows from (1.9) that:

(1.10)

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

Definition 1Denote 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 2A 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 1The function$L z , t :U× 0 ; 1 →ℂ$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 2Suppose that the function$H: ℂ 2 →ℂ$satisfies the condition

$Re { H ( i s ; t ) } ≤0$

for all real s and for all t-n (1 + s2) / 2, $n∈ℕ$. If the function p(z) = 1+p n zn +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 3Let κ, $γ∈ℂ$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 4Let$p∈F a$and let q(z) = a + a n zn + 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

$q ( U r 0 ) ⊂p ( U ) ;q ( z 0 ) =p ( ζ 0 ) and z 0 p ′ ( z 0 ) =m ζ 0 p ′ ( ζ 0 ) ( m ≥ n ) .$

Lemma 5Let q H[a; 1] and $ϕ: ℂ 2 →ℂ$. 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 $b ∈ ℂ \ ℤ 0 -$, $s∈ℂ$, Re {b}, µ > 0, $p∈ℕ$, $z∈U$ 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

$Re { H ( i ϑ ; t ) } ≤ 0 ϑ ∈ ℝ ; t ≤ - 1 + ϑ 2 2 ,$
(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

$c 1 -2δ≥0,$

or

$c 1 ≥2δ= 1 + | c | 2 - | 1 - c 2 | 2 c 1 .$

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

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

$F ( z 0 ) =G ( ζ 0 ) and z 0 F ′ ( z 0 ) = ( 1 + t ) ζ 0 G ′ ( ζ 0 ) ( 0 ≤ t < ∞ ) .$
(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, $p∈ℕ$) in our results of this article, we obtain the results obtained by Aouf and Seoudy;

(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.

## References

1. 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.

2. 2.

Miller S, Mocanu PT: Subordinants of differential superordinations. Complex Var Theory Appl 2003, 48(10):815–826. 10.1080/02781070310001599322

3. 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.2004.12.004

4. 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-X

5. 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. 10.1080/10652460600926923

6. 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.2005.01.020

7. 7.

Srivastava HM, Choi J: Series Associated with the Zeta and Related Functions. Kluwer Academic Publishers, Dordrecht; 2001.

8. 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. 10.1080/10652460701208577

9. 9.

Alexander JW: Functions which map the interior of the unit circle upon simple regions. Ann Math Ser 1915, 17(2):12–22.

10. 10.

Libera RJ: Some classes of regular univalent functions. Proc Am Math Soc 1969, 16: 755–758.

11. 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.1993.1204

12. 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. 10.1080/10652460701208577

13. 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)00500-0

14. 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)

15. 15.

El-Ashwah RM, Aouf MK: Some properties of new integral operator. Acta Univ Apulensis 2010, 24: 51–61 (2010).

16. 16.

Pommerenke Ch: Univalent Functions. Vandenhoeck and Ruprecht, Göttingen 1975.

17. 17.

Miller SS, Mocanu PT: Differential subordinations and univalent functions. Michigan Math J 1981, 28(2):157–172.

18. 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)90082-8

19. 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.

## Author information

Authors

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors read and approved the final manuscript.

## Rights and permissions

Reprints and Permissions

Aouf, M.K., Mostafa, A.O., Shahin, A.M. et al. Some subordination and superordination results of generalized Srivastava-Attiya operator. J Inequal Appl 2012, 115 (2012). https://doi.org/10.1186/1029-242X-2012-115 