# Convexity of Certain Integral Operators Defined by Struve Functions

Authors: Shahid Mahmood; Sana Mahroz; Ayesha Rafiq; Sarfraz Nawaz Malik; Mohsan Raza
Journal: Journal of Function Spaces
 (2018)

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## Abstract

This article deals with some functional inequalities involving Struve function, generalized Struve function, and modified Struve functions. We aim to find the convexity of the integral operator defined by Struve function, generalized Struve function, and modified Struve functions.


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## Body

## 1. Introduction

We denote byA the class of functions f which are analytic in the open unit disc U=z:z<1 and of the form(1)fz=z+∑n=2∞anzn.Let S denote the class of all functions in A which are univalent in U. Also let S∗ and C be the subclasses of S consisting of all functions which map U onto a star shaped with respect to origin and convex domains, respectively. These classes are defined as (2)S∗=f:f∈A,Rezf′zfz>0,z∈U,C=f:f∈A,Re1+zf′′zf′z>0,z∈U. Ozaki [1] showed that if a function f is analytic in U and of the form (1) such that f(z)f′(z)/z≠0 and if either (3)R1+zf′′zf′z≥-12,z∈U or (4)R1+zf′′zf′z≤32,z∈U, then f is univalent and convex in at least one direction in U. It shows that the constants -1/2 and 3/2 are, in a certain sense, the best possible constants. In [2], for ∈R, consider the class G(β) consisting of locally univalent functions f∈A that satisfy the condition R1+zf′′z/f′z<1+β/2 for 0<β≤1. That is, (5)Gβ=f∈A:R1+zf′′zf′z<1+β2,0<β≤1. It can be seen that Gβ≠⌀ if and only if β>0. Since Gβ1⊂Gβ2 whenever β1<β2, it readily follows that the class Gβ is in S∗, for β∈(0,1], so in particular, the functions in Gβ are univalent functions. For some more detail about these conditions, see [3, 4].Now we consider the second-order inhomogeneous differential equation(6)z2w′′z+zw′z-z2+v2wz=4z/2v+1πΓv+1/2. The solution of the homogeneous part is Bessel functions of order v, where v is real or complex number. For some more details, see [5]. The particular solution of the inhomogeneous equation defined in (6) is called the Struve function of order v. It is defined as(7)Kvz=∑n=0∞-1nz/22n+v+1Γn+3/2Γv+n+3/2.Now we consider the differential equation(8)z2w′′z+zw′z-z2+v2wz=4z/2v+1πΓv+1/2.Equation (8) differs from (6) in the coefficients of w. Its particular solution is called the modified Struve functions of order v and is given by(9)Lvz=-ie-ivπ/2Kviz=∑n=0∞z/22n+v+1Γn+3/2Γv+n+3/2.Again consider the second-order in-homogenous differential equation(10)z2w′′z+bzw′z+cz2-v2+1-bvwz=4z/2v+1πΓv+b/2,where b,c,v∈C. Equation (10) generalizes (6) and (8). In particular for b=1,c=1, we obtain (6) and for b=1,c=-1, we obtain (8). Its particular solution has the series form(11)wv,b,cz=∑n=0∞-1ncnz/22n+v+1Γn+3/2Γv+n+b+2/2 and is called the generalized Struve function of order v. This series is convergent everywhere. We take the transformation (12)uv,b,cz=2vπΓv+b+2/2z-v-1/2wv,b,cz=∑n=0∞-c/4nzn3/2nln,where l=v+b+2/2≠0,-1,-2,… and γn=Γγ+n/Γγ=γγ+1⋯γ+n-1, where Γ· denotes the Gamma function. The function uv,b,c is analytic in the whole complex plane and satisfies the differential equation (13)4z2w′′z+22p+b+3zw′z+cz+2p+bwz=2p+b. The function uv,b,c unifies the Struve functions and modified Struve functions.The functionuv,b,c was introduced and studied by Orhan and Yagmur [6] and further investigated by [7–9]. In last few years, many mathematicians have set the univalence criteria of several those integral operators which preserve the class S. By using a variety of different analytic techniques, operators, and special functions, several authors have studied univalence criterion. Recently Din et al. [10] studied the univalence of integral operators involving generalized Struve functions. These operators are defined as follows:(14)Fα1,…,αn,βz=β∫0ztβ-1∏i=1nvli,b,ctt1/αidt1/β,Xn,γz=nγ+1∫0z∏i=1nvli,b,ctγdt1/nγ+1,Zλz=λ∫0Ztλ-1evli,b,ctλdt1/λ.In this paper, our main aim is to study the convexity and univalence of the integral operators(15)Wvi,b,c,γiz=∫0z∏i=1nuvi,b,cttγidt,(16)Gvi,b,c,γiz=∫0z∏i=1nuvi,b,c′tγidt.Applications of Struve functions occur in water wave and surface wave problems, unsteady aerodynamics, resistive MHD instability theory, and optical diffraction. More recently, Struve functions appeared in particle quantum dynamic studies of spin decoherence and nanotubes electrodynamics, potential theory, and optics. For more details, we refer to see [11].Lemma 1 (see [6]).
Ifb,v∈R and c∈C, l=v+b+2/2 are so constrained that l>max⁡0,7c/24, then the function uv,b,c:U→C satisfies the following inequalities:(i)
zuv,b,c′z/uv,b,cz-1≤|c|(6l-|c|)/3(4l-|c|)(3l-|c|),(ii)
zuv,b,c′′z/uv,b,c′z≤6c/12l-7c.


## 2. Main Results

In this section, we find the convexity of these integral operators defined by generalized Struve functions, by using above lemma and the following inequalities. We also use Ozaki’s condition for the univalence of these operators.


### 2.1. Convexity Criteria for Integral Operators

Theorem 2.
Letv1,…,vn,b∈R and c∈C be so constrained that vi>7c/24, where li=vi+b+2/2(i=1,2,3,…,n). Consider the function uvi,b,c:U→C, defined as (17)uvi,b,cz=2viπΓvi+b+22z-1-vi/2wvi,b,cz. Also suppose that l=min⁡l1,l2,…,ln and γ1,γ2,…,γn be positive real number and that satisfies the inequality (18)0≤1-c6l-c34l-c3l-c∑i=1nγi<1, then Wvi,b,c,γiz:U→C defined in (15) is in class Cα and (19)α=1-c6l-c34l-c3l-c∑i=1nγi.Proof.
It is clear thatuvi,b,c∈A as it satisfies the condition (20)uvi,b,c0=uvi,b,c′-1=0. Differentiating (15), we obtain(21)Wvi,b,c,γi′z=∏i=1nuvi,b,czzγi.Clearly (22)Wvi,b,c,γi0=Wvi,b,c,γi′0-1=0. Differentiating logarithmically and after simple calculations, it follows that(23)R1+zWvi,b,c,ri′′zWvi,b,c,γi′z=∑i=1nγiRzuvi,b,c′zuvi,b,cz+∑i=1nγi.Using inequality (i) of Lemma 1, it is easy to see that (24)zuv,b,c′zuv,b,cz-1≤c6l-c34l-c3l-c, so (25)Rzuv,b,c′zuv,b,cz>1-c6l-c34l-c3l-c. By putting the above inequality in (23), we have (26)R1+zWvi,b,c,ri′′zWvi,b,c,γi′z>∑i=1nγi1-c6li-c34li-c3li-c+1-∑i=1nγi=∑i=1nγi-∑i=1nγic6li-c34li-c3li-c+1-∑i=1nγi>1-∑i=1nγic6li-c34li-c3li-c. For z∈U and li=vi+b+2/2>7c/24,∀i=1,…,n, we consider that the function (27)Ω:7c24,∞⟶R, defined by (28)Ωx=c6x-c34x-c3x-c, is decreasing ∀i=1,…,n, and therefore (29)c6li-c34li-c3li-c≤c6l-c34l-c3l-c. Hence (30)R1+zWvi,b,c,ri′′zWvi,b,c,γi′z>1-c6l-c34l-c3l-c∑i=1nγi, which completes the proof that Wvi,b,c,γiz∈Cα.Theorem 3.
Letv1,…,vn,b∈R and c∈C be so constrained that li>7c/24 where li=vi+b+2/2(i=1,2,3,…,n). Consider the function uvi,b,c:U→C, defined as (31)uvi,b,cz=2viπΓvi+b+22z-1-vi/2wvi,b,cz. Also suppose that l=min⁡l1,l2,…,ln and γ1,γ2,…,γn be positive real number and that satisfies the inequality (32)0≤1-6c12l-7c∑i=1nγi<1, then Gvi,b,c,γiz:U→C defined in (16) is in class Cα, where (33)α=1-6c12l-7c∑i=1nγi.Proof.
Differentiating (16), we obtain(34)Gvi,b,c,γi′z=∏i=1nuvi,b,c′zγi.Clearly (35)Gvi,b,c,γi0=Gvi,b,c,γi′0-1=0. Differentiating logarithmically and after simple calculations, we obtain(36)R1+zGvi,b,c,ri′′zGvi,b,c,γi′z=∑i=1nγiRzlvi,b,c′′zlvi,b,c′z+1.By using inequality (ii) of Lemma 1, we have (37)zuvi,b,c′′zuvi,b,c′z≤6c12l-7c, so (38)Rzuvi,b,c′′zuvi,b,c′z>-6c12l-7c. Using above relation in (36), we have (39)R1+zGvi,b,c,ri′′zGvi,b,c,γi′z>∑i=1nγi-6c12li-7c+1=1-∑i=1nγi6c12li-7c. For z∈U and li=vi+b+2/2>7c/24i=1,…,n,we consider that the function (40)Ψ:7c24,∞⟶R, defined by (41)Ψx=6c12x-7c, is decreasing ∀i=1,…,n, and therefore (42)6c12li-7c≤6c12l-7c. Hence (43)R1+zGvi,b,c,ri′′zGvi,b,c,γi′z>1-∑i=1nγi6c12l-7c. We have α=1-6c/12l-7c∑i=1nγi which completes the proof that Gvi,b,c,γiz∈Cα.


## 2.1. Convexity Criteria for Integral Operators

Theorem 2.
Letv1,…,vn,b∈R and c∈C be so constrained that vi>7c/24, where li=vi+b+2/2(i=1,2,3,…,n). Consider the function uvi,b,c:U→C, defined as (17)uvi,b,cz=2viπΓvi+b+22z-1-vi/2wvi,b,cz. Also suppose that l=min⁡l1,l2,…,ln and γ1,γ2,…,γn be positive real number and that satisfies the inequality (18)0≤1-c6l-c34l-c3l-c∑i=1nγi<1, then Wvi,b,c,γiz:U→C defined in (15) is in class Cα and (19)α=1-c6l-c34l-c3l-c∑i=1nγi.Proof.
It is clear thatuvi,b,c∈A as it satisfies the condition (20)uvi,b,c0=uvi,b,c′-1=0. Differentiating (15), we obtain(21)Wvi,b,c,γi′z=∏i=1nuvi,b,czzγi.Clearly (22)Wvi,b,c,γi0=Wvi,b,c,γi′0-1=0. Differentiating logarithmically and after simple calculations, it follows that(23)R1+zWvi,b,c,ri′′zWvi,b,c,γi′z=∑i=1nγiRzuvi,b,c′zuvi,b,cz+∑i=1nγi.Using inequality (i) of Lemma 1, it is easy to see that (24)zuv,b,c′zuv,b,cz-1≤c6l-c34l-c3l-c, so (25)Rzuv,b,c′zuv,b,cz>1-c6l-c34l-c3l-c. By putting the above inequality in (23), we have (26)R1+zWvi,b,c,ri′′zWvi,b,c,γi′z>∑i=1nγi1-c6li-c34li-c3li-c+1-∑i=1nγi=∑i=1nγi-∑i=1nγic6li-c34li-c3li-c+1-∑i=1nγi>1-∑i=1nγic6li-c34li-c3li-c. For z∈U and li=vi+b+2/2>7c/24,∀i=1,…,n, we consider that the function (27)Ω:7c24,∞⟶R, defined by (28)Ωx=c6x-c34x-c3x-c, is decreasing ∀i=1,…,n, and therefore (29)c6li-c34li-c3li-c≤c6l-c34l-c3l-c. Hence (30)R1+zWvi,b,c,ri′′zWvi,b,c,γi′z>1-c6l-c34l-c3l-c∑i=1nγi, which completes the proof that Wvi,b,c,γiz∈Cα.Theorem 3.
Letv1,…,vn,b∈R and c∈C be so constrained that li>7c/24 where li=vi+b+2/2(i=1,2,3,…,n). Consider the function uvi,b,c:U→C, defined as (31)uvi,b,cz=2viπΓvi+b+22z-1-vi/2wvi,b,cz. Also suppose that l=min⁡l1,l2,…,ln and γ1,γ2,…,γn be positive real number and that satisfies the inequality (32)0≤1-6c12l-7c∑i=1nγi<1, then Gvi,b,c,γiz:U→C defined in (16) is in class Cα, where (33)α=1-6c12l-7c∑i=1nγi.Proof.
Differentiating (16), we obtain(34)Gvi,b,c,γi′z=∏i=1nuvi,b,c′zγi.Clearly (35)Gvi,b,c,γi0=Gvi,b,c,γi′0-1=0. Differentiating logarithmically and after simple calculations, we obtain(36)R1+zGvi,b,c,ri′′zGvi,b,c,γi′z=∑i=1nγiRzlvi,b,c′′zlvi,b,c′z+1.By using inequality (ii) of Lemma 1, we have (37)zuvi,b,c′′zuvi,b,c′z≤6c12l-7c, so (38)Rzuvi,b,c′′zuvi,b,c′z>-6c12l-7c. Using above relation in (36), we have (39)R1+zGvi,b,c,ri′′zGvi,b,c,γi′z>∑i=1nγi-6c12li-7c+1=1-∑i=1nγi6c12li-7c. For z∈U and li=vi+b+2/2>7c/24i=1,…,n,we consider that the function (40)Ψ:7c24,∞⟶R, defined by (41)Ψx=6c12x-7c, is decreasing ∀i=1,…,n, and therefore (42)6c12li-7c≤6c12l-7c. Hence (43)R1+zGvi,b,c,ri′′zGvi,b,c,γi′z>1-∑i=1nγi6c12l-7c. We have α=1-6c/12l-7c∑i=1nγi which completes the proof that Gvi,b,c,γiz∈Cα.


## 3. Some Special Cases of Struve Functions and Modified Struve Functions

### 3.1. Struve Functions

By settingb=c=1 in (11), we obtain the Struve function of first kind of order v, denoted by Kv(z) defined in (7). Let Kv:U→C be defined as (44)Kvz=2vπΓv+b+22z-1-v/2Kvz. We observe that (45)K-1/2z=zsin⁡z,K1/2z=21-cos⁡z,K3/2z=41+2z-8sin⁡zz+cos⁡zz.Corollary 4.
Letv1,…,vn>-1.75, where n∈N. Consider the function Kvz:U→C defined by(46)Kviz=2viπΓvi+32z-vi-1/2Kviz.Also let v=min⁡v1,v2,…,vn and γ1,γ2,…,γn be positive real numbers. Suppose that these numbers satisfy the following inequality.(47)0≤1-43v+4324v2+58v+35∑i=1nγi<1,then the function Wvi,γiz:U→C defined as(48)Wvi,γiz=∫0z∏i=1nKvittγidtis in class Cα, where (49)α=1-43v+4324v2+58v+35∑i=1nγi. In particular
(i) First atv=-1/2, the function(50)W-1/2,γz=∫0zK-1/2ttγdt=∫0zsin⁡ttγdtis in class Cα1 for γ≤18/5, where (51)α1=1-518γ.
(ii) Forv=1/2, then the function (52)W1/2,γz=∫0zK1/2ttγdt=∫0z21-costtγdtis in class Cα2 for γ≤105/11, where α2=1-11/105γ.
(iii) Forv=3/2, the function(53)W3/2,γz=∫0zK3/2ttγdt=∫0z41+2/t-8sin⁡t/t+cos⁡t/ttγdtis in class Cα3 for γ≤264/17, where α3=1-17/264γ.Corollary 5.
Letv1,…,vn>-1.75 where n∈N. Consider the function Kvz:U→C defined by (46) and v=min⁡v1,v2,…,vn and γ1,γ2,…,γn be positive real numbers such that these numbers satisfy the inequality(54)0≤1-612v+11∑i=1nγi<1,then the function Gvi,γiz:U→C defined as(55)Gvi,γiz=∫0z∏i=1nKti′tγidtis in class Cα, where 0≤α<1 and (56)α=1-612v+11∑i=1nγi. In particular, one has the following:
(i) Forv=-1/2, then(57)G-1/2,γz=∫0zK-1/2′tγdt=∫0z12tsin⁡t+cos⁡t2γdtis in class Cα1 for γ≤5/6, where α1=1-6/5γ.
(ii) Forv=1/2, the function(58)G1/2,γz=∫0zK1/2′tγdtG1/2,γz=∫0zsin⁡ttγdtis in class Cα2 for γ≤17/6, where α2=1-6/17γ.
(iii) Forv=3/2, the function(59)G3/2,γz=∫0zK3/2′tγdt=∫0z-8t2-8cos⁡t2t-sin⁡t2tt-sin⁡t2t-cos⁡t2ttγdt,
is in classCα3 for γ≤29/6, where α3=1-6/29γ.


### 3.2. Modified Struve Functions

We obtain the modified Struve function of first kind of orderv, denoted by Lv(z), defined by (9), by putting b=-c=1 in (11). Define a function Lv(z):U→C by (60)Lvz=2vπΓv+32z-1-v/2Lvz. We observe that (61)L1/2z=2cosh⁡z-1. By making use of this particular value, we have the following assertions.Corollary 6.
Letv1,…,vn>-1.75 where n∈N. Consider the function Lvz:U→C defined by (62)Lviz=2viπΓvi+32z-vi-1/2Lviz, and v=min⁡v1,v2,…,vn and γ1,γ2,…,γn be positive real numbers. Suppose that these numbers satisfy the inequality (63)0≤1-43v+4324v2+58v+35∑i=1nγi<1, then the function Wvi,γiz:U→C defined as(64)Wvi,γiz=∫0z∏i=1nLvittγidt,is in class Cα, where (65)α=1-43v+4324v2+58v+35∑i=1nγi.
(i) In particular forv=1/2, the integral operator(66)W1/2,γz=∫0zL1/2ttγdt=∫0z2cosh⁡t-1tγdtis in class Cα1 for γ≤105/11, where α1=1-11/105γ.Corollary 7.
Letv1,…,vn>-1.75, where n∈N. Consider the function Lvz:U→C defined by (67)Lviz=2viπΓvi+32z-vi-1/2Lviz,v=min⁡v1,v2,…,vn, and γ1,γ2,…,γn be positive real numbers. Suppose that these numbers satisfy the relation (68)0≤1-612v+11∑i=1nγi<1, then the function Gvi,γiz:U→C defined as (69)Gvi,γiz=∫0z∏i=1nLvi′tγidt is Cα, where (70)α=1-612v+11∑i=1nγi. In particular for v=1/2, the function(71)G1/2,γz=∫0zL1/2′tγdt=∫0zsinh⁡ttγdtis in class Cα1 for γ≤17/6, where α1=1-6/17γ.


## 3.1. Struve Functions

By settingb=c=1 in (11), we obtain the Struve function of first kind of order v, denoted by Kv(z) defined in (7). Let Kv:U→C be defined as (44)Kvz=2vπΓv+b+22z-1-v/2Kvz. We observe that (45)K-1/2z=zsin⁡z,K1/2z=21-cos⁡z,K3/2z=41+2z-8sin⁡zz+cos⁡zz.Corollary 4.
Letv1,…,vn>-1.75, where n∈N. Consider the function Kvz:U→C defined by(46)Kviz=2viπΓvi+32z-vi-1/2Kviz.Also let v=min⁡v1,v2,…,vn and γ1,γ2,…,γn be positive real numbers. Suppose that these numbers satisfy the following inequality.(47)0≤1-43v+4324v2+58v+35∑i=1nγi<1,then the function Wvi,γiz:U→C defined as(48)Wvi,γiz=∫0z∏i=1nKvittγidtis in class Cα, where (49)α=1-43v+4324v2+58v+35∑i=1nγi. In particular
(i) First atv=-1/2, the function(50)W-1/2,γz=∫0zK-1/2ttγdt=∫0zsin⁡ttγdtis in class Cα1 for γ≤18/5, where (51)α1=1-518γ.
(ii) Forv=1/2, then the function (52)W1/2,γz=∫0zK1/2ttγdt=∫0z21-costtγdtis in class Cα2 for γ≤105/11, where α2=1-11/105γ.
(iii) Forv=3/2, the function(53)W3/2,γz=∫0zK3/2ttγdt=∫0z41+2/t-8sin⁡t/t+cos⁡t/ttγdtis in class Cα3 for γ≤264/17, where α3=1-17/264γ.Corollary 5.
Letv1,…,vn>-1.75 where n∈N. Consider the function Kvz:U→C defined by (46) and v=min⁡v1,v2,…,vn and γ1,γ2,…,γn be positive real numbers such that these numbers satisfy the inequality(54)0≤1-612v+11∑i=1nγi<1,then the function Gvi,γiz:U→C defined as(55)Gvi,γiz=∫0z∏i=1nKti′tγidtis in class Cα, where 0≤α<1 and (56)α=1-612v+11∑i=1nγi. In particular, one has the following:
(i) Forv=-1/2, then(57)G-1/2,γz=∫0zK-1/2′tγdt=∫0z12tsin⁡t+cos⁡t2γdtis in class Cα1 for γ≤5/6, where α1=1-6/5γ.
(ii) Forv=1/2, the function(58)G1/2,γz=∫0zK1/2′tγdtG1/2,γz=∫0zsin⁡ttγdtis in class Cα2 for γ≤17/6, where α2=1-6/17γ.
(iii) Forv=3/2, the function(59)G3/2,γz=∫0zK3/2′tγdt=∫0z-8t2-8cos⁡t2t-sin⁡t2tt-sin⁡t2t-cos⁡t2ttγdt,
is in classCα3 for γ≤29/6, where α3=1-6/29γ.


## 3.2. Modified Struve Functions

We obtain the modified Struve function of first kind of orderv, denoted by Lv(z), defined by (9), by putting b=-c=1 in (11). Define a function Lv(z):U→C by (60)Lvz=2vπΓv+32z-1-v/2Lvz. We observe that (61)L1/2z=2cosh⁡z-1. By making use of this particular value, we have the following assertions.Corollary 6.
Letv1,…,vn>-1.75 where n∈N. Consider the function Lvz:U→C defined by (62)Lviz=2viπΓvi+32z-vi-1/2Lviz, and v=min⁡v1,v2,…,vn and γ1,γ2,…,γn be positive real numbers. Suppose that these numbers satisfy the inequality (63)0≤1-43v+4324v2+58v+35∑i=1nγi<1, then the function Wvi,γiz:U→C defined as(64)Wvi,γiz=∫0z∏i=1nLvittγidt,is in class Cα, where (65)α=1-43v+4324v2+58v+35∑i=1nγi.
(i) In particular forv=1/2, the integral operator(66)W1/2,γz=∫0zL1/2ttγdt=∫0z2cosh⁡t-1tγdtis in class Cα1 for γ≤105/11, where α1=1-11/105γ.Corollary 7.
Letv1,…,vn>-1.75, where n∈N. Consider the function Lvz:U→C defined by (67)Lviz=2viπΓvi+32z-vi-1/2Lviz,v=min⁡v1,v2,…,vn, and γ1,γ2,…,γn be positive real numbers. Suppose that these numbers satisfy the relation (68)0≤1-612v+11∑i=1nγi<1, then the function Gvi,γiz:U→C defined as (69)Gvi,γiz=∫0z∏i=1nLvi′tγidt is Cα, where (70)α=1-612v+11∑i=1nγi. In particular for v=1/2, the function(71)G1/2,γz=∫0zL1/2′tγdt=∫0zsinh⁡ttγdtis in class Cα1 for γ≤17/6, where α1=1-6/17γ.


## 4. Locally Univalence Criteria

Now, we find the locally univalent criteria of the integral operators involving generalized Struve functions defined in (15), (16).Theorem 8.
Letv1,…,vn,b∈R and c∈C such that li>7c/24, where li=vi+b+2/2(i=1,…,n). Consider the function uvi,b,c:U→C, defined as (72)uvi,b,cz=2viπΓvi+b+22z-1-vi/2wvi,b,cz. Also suppose that l=min⁡l1,l2,…,ln and γ1,γ2,…,γn be positive real number and that satisfies the inequality (73)0<8c12l-7c∑i=1nγi≤1, then Gvi,b,c,γiz:U→C defined in (16) is in class Gβ, where (74)β=8c12l-7c∑i=1nγi.Proof.
Differentiating (16), we obtain (75)Gvi,b,c,γiz=∏i=1nuvi,b,c′zγi. Differentiating logarithmically, we obtain(76)R1+zGvi,b,c,γi′′zGvi,b,c,γi′z=∑i=1nγiRzuvi,b,c′′zuvi,b,c′z+1.By using (ii) of Lemma 1, it follows that (77)Rzuv,b,c′′zuv,b,c′z≤6c12l-7c. Therefore, we have(78)R1+zGvi,b,c,γi′′zGvi,b,c,γi′z≤∑i=1nγi6c12li-7c+1<1+2∑i=1nγi6c/12li-7c2<1+∑i=1nγi8c/12li-7c2.For z∈U and li>7c/24∀i=1,…,n, we consider the function (79)Φ:7c24,∞⟶R, defined by (80)Φx=128c12x-7c, is decreasing ∀i=1,…,n; therefore (81)128c12li-7c≤128c12l-7c. This implies that (82)R1+zGvi,b,c,γi′′zGvi,b,c,γi′z<1+128c12l-7c∑i=1nγi. Let (83)0≤8c12l-7c∑i=1nγi<1. Hence (84)R1+zGti,a,b,ri′′zGti,a,b,γi′z<1+β2. Here β=8c/12l-7c∑i=1nγi, which completes the proof.Theorem 9.
Letv1,…,vn,b∈R and c∈C be such that li>7c/24, where li=vi+b+2/2(i=1,…,n). Consider the function uvi,a,b:U→C, defined as (85)uvi,b,cz=2viπΓvi+b+22z-1-vi/2wvi,b,cz. Also suppose that l=min⁡l1,l2,…,ln and γ1,γ2,…,γn are positive real number and that satisfies the inequality (86)0<2c6l-c34l-c3l-c∑i=1nγi≤1, then Wvi,b,c,γiz:U→C defined in (15) is in class Gβ, where (87)β=2c6l-c34l-c3l-c∑i=1nγi.Proof.
Differentiating (15), we have(88)Wvi,b,c,γi′z=∑i=1nuvi,b,czzγi.Differentiating logarithmically, we obtain(89)R1+zWvi,b,c,γi′′zWvi,b,c,γi′z=∑i=1nγiRzuvi,b,c′zuvi,b,cz+1-∑i=1nγi.Now consider (i) of Lemma 1, we obtain (90)Rzuv,b,c′zuv,b,cz<1+c6l-c34l-c3l-c. Therefore (91)R1+zWvi,b,c,γi′′zWvi,b,c,γi′z≤∑i=1nγi1+c6li-c34li-c3li-c+1-∑i=1nγi. For z∈U and li>7c/24∀i=1,…,n, we consider the function (92)Λ:7c24,∞⟶R, defined by (93)Λx=122c6x-c34x-c3x-c; therefore (94)12∑i=1nγi2c6li-c34li-c3li-c≤12i2c6l-c34l-c3l-c∑i=1nγi. This implies that (95)R1+zWvi,b,c,γi′′zWvi,b,c,γi′z<1+2c6l-c/34l-c3l-c∑i=1nγi2. Let (96)0≤2c6l-c34l-c3l-c∑i=1nγi<1. Hence (97)R1+zWvi,b,c,γi′′zWvi,b,c,γi′z<1+β2. Here β=2c6l-c/34l-c3l-c∑i=1nγi, which completes the proof.



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