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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 : | 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 = { 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 bythe set of all functions q(z) that are analytic and injective onŪ\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 byF a , F 0 F 0 andF 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 functionL 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 functionH: 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 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 3[18]Let κ, γwith 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]LetpF 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 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 andqH 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 solutionqF 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, zU 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 -2δ0,

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

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 ) 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) 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.

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Correspondence to Samar Mohamed Madian.

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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

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Keywords

  • analytic function
  • Hadamard product
  • differential subordination
  • superordination