# Certain Analytic Functions Defined by Generalized Mittag-Leffler Function Associated with Conic Domain

Authors: Adel A. Attiya; T. M. Seoudy; M. K. Aouf; Abeer M. Albalahi
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In the present paper, we investigate and introduce several properties of certain families of analytic functions in the open unit disc, which are defined byq-analogue of Mittag-Leffler function associated with conic domain. A number of coefficient estimates of the functions in these classes have been obtained. Sufficient conditions for the functions belong to these classes are also considered.


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## Body

## 1. Introduction

Denoted byA, the class of all analytic functions in D=ζ∈ℂ:ζ<1 has the following form:
(1)gζ=ζ+∑m=2∞bmζm.For two functionsg1ζ and g2ζ, analytic in D, we say that the function g1ζ is subordinate to g2ζ, written as g1ζ≺g2ζ, if there exists a Schwarz function wζ, which is analytic in D with w0=0 and wζ<1 such that g1ζ=g2wζ. Furthermore, if the function g2ζ is univalent in D, then we have the following equivalence:
(2)g1z≺g2z⟺g10=g20,g1D⊂g2D.LetS be the subclass of A consisting of univalent functions. Also, let S∗η and Cη denote the subclasses of univalent starlike and convex functions of order η (0≤η<1). A function g∈S is said to be k-starlike function, written as g∈STk, if
(3)Rζg′ζgζ>kζg′ζgζ−10≤k<∞;ζ∈D.A functiong∈S is said to be k-convex function, written as g∈UCVk, if
(4)Rζgζ′g′ζ>kζgζ′g′ζ−10≤k<∞;ζ∈D.The classesk−ST and k−UCV were introduced and studied by Kanas and Wiśniowska [1, 2] (see also, [3–8]). In particular, when k=1, we get ST1≡ST and UCV1≡UCV, where ST and UCV are the familiar classes of uniformly starlike functions and uniformly convex functions in D, respectively (see [9]). A function g∈A is said to be in the class STk,η that are k-starlike functions of order η with 0≤η<1 if
(5)Rζg′ζgζ−η>kζg′ζgζ−10≤k<∞;ζ∈D.A functiong∈A is said to be in the class UCVk,η that are k-convex functions of order η with 0≤η<1 if
(6)Rζgζ′g′ζ−η>kζgζ′g′ζ−10≤k<∞;ζ∈D.The classesSTk,η and UCVk,η have been studied by Kanas and Ra˘ducanu [10]. We note that STk,0=k−ST and UCVk,0=k−UCV. The classes ST1,η≡STη and UCV1,η≡UCVη were investigated in [11–13].The study of quantum calculus (orq-calculus) attracted the researches due to its applications in various branches of mathematics and physics, for example, in the areas of special functions, q-difference, ordinary fractional calculus, q-integral equations, and in q-transform analysis (see [14–23]).For0<q<1 and g∈A given by (1), the q-derivative of g is defined by (see [24–29]):
(7)Dqgζ=g′0ifζ=0,gζ−gqζ1−qζifζ≠0,provided that g′0 exists. From (1) and (7), we have
(8)Dqgζ=1+∑m=2∞mqbmζm−1ζ≠0,where mq is q-integer number m defined by
(9)mq=1−qm1−q=1+q+q2+⋯+qm−10<q<1.We note that(10)limq⟶1−Dqgζ=limq⟶1−gζ−gqζ1−qζ=g′ζ,for a function g which is differentiable in D. Making use of the q-derivative operator Dq given by (7), we introduce the subclasses STqk,η and UCVqk,η in D as follows.Definition 1.
For0<q<1, 0≤η<1, 0≤k<∞, and ζ∈D, let STqk,η and UCVqk,η be the subclasses of A consisting of functions g of the form (1) and satisfy the analytic criterion:
(11)RζDqgζgζ−η>kζDqgζgζ−1,(12)RDqζDqgζDqgζ−η>kDqζDqgζDqgζ−1.From (11) and (12), it follows that
(13)g∈UCVqk,η⟺ζDqg∈STqk,η.Theq-shifted factorials, for any complex number α, are defined by
(14)α;qn=1n=0,∏k=0n−11−αqkn∈ℕ≪1,2,3,⋯.The definition (14) remains meaningful for n=∞ as a convergent infinite product
(15)α;q∞=∏j=0∞1−αqjforq<1.Furthermore, in terms of theq-gamma function Γqζ defined by
(16)Γqζ≔q;q∞1−q1−ζqζ;q∞0<q<1;ζ∈ℂ,so that
(17)limq⟶1−Γqζ=Γζ,for the familiar gamma function Γζ, we find from (14) that
(18)qα;qn=Γqα+nΓqα1−qnn∈ℕ;α∈ℂ.We note that(19)limq⟶1−qα;qn1−qn=αn,where
(20)αn=Γα+nΓα=1n=0,αα+1⋯α+n−1n∈ℕ.For0<q<1, α,β,γ∈ℂ, Rα>0, Rβ>0, and Rγ>0, consider the q-analogue of Mittag-Leffler defined by Sharma and Jain [30], for generalized Mittag-Leffler function (see, e.g., [31–33])
(21)Eα,βγζ;q=∑m=0∞qγ;qmq;qmζmΓqαm+β.Asq⟶1−, the operator Eα,βγζ;q reduces to Eα,βγζ introduced by Prabhakar [34]. Now, let us define
(22)Eα,βγζ;q≔ζΓqβEα,βγζ;q=ζ+∑m=2∞qγ;qm−1q;qm−1ΓqβΓqαm−1+βζm.We remark that:(i)
E1,11ζ;q=ζeqζ(ii)
E1,21ζ;q=eqζ−1where eqζ is one of the q-analogues of the exponential function eζ given by
(23)eqζ=∑m=0∞ζmΓqm+1=∑m=0∞1−qmζmq;qm=1ζ;q∞.Using the Hadamard product (or convolution), we define the linear operatorHq,α,βγ:A⟶A by
(24)Hq,α,βγgζ=Eα,βγζ;q∗gζ=ζ+∑m=2∞qγ;qm−1q;qm−1ΓqβΓqαm−1+βbmζmζ∈D.We note that (see [35, 36])
(25)limq⟶1−Hq,α,βγgζ=Hα,βγgζ=ζ+∑m=2∞γm−1ΓβΓαm−1+βm−1!bmζmζ∈D.Motivated by the works of Kanas and Yaguchi [37] and Kanas and Ra˘ducanu [10], we define the following classes of functions with the theory of q-calculus.Definition 2.
For0<q<1, α,β,γ∈ℂ, Rα>0, Rβ>0, Rγ>0, 0≤k<∞, and 0≤η<1, let
(26)STq,α,βγk,η=g∈A:Hq,α,βγg∈STqk,η,UCVq,α,βγk,η=g∈A:Hq,α,βγg∈UCVqk,η.
It is easy to check that(27)g∈UCVq,α,βγk,η⟺ζDqg∈STq,α,βγk,η.Takingq⟶1− in Definition 2, we obtain
(28)limq⟶1−STq,α,βγk,η=STα,βγk,η=g∈A:RζHα,βγgζ′Hα,βγgζ−η>kζHα,βγgζ′Hα,βγgζ−1,STq,0,β1k,η=STqk,η,limq⟶1−STq,0,β1k,η=STk,η,(29)limq⟶1−UCVq,α,βγk,η=UCVα,βγk,η=g∈A:RζHα,βγgζ′′Hα,βγgζ′−η>kζHα,βγgζ′′Hα,βγgζ′−1,UCVq,0,β1k,η=UCVqk,η,limq⟶1−UCVq,0,β1k,η=UCVk,η.Motivated by the works mentioned above, in this paper, we will investigate some important properties, coefficient estimates, and the familiar Fekete–Szegö type inequalities for the subclassesSTq,α,βγk,η and UCVq,α,βγk,η.


## 2. Some Results of Functions inSTq,α,βγk,η and UCVq,α,βγk,η

Unless otherwise mentioned, we assume throughout this paper that0<q<1, α,β,γ>0, 0≤k<∞, 0≤η<1, and ζ∈D.Letg∈STq,α,βγk,η, we have
(30)RζDqHq,α,βγgζHq,α,βγgζ−η>kζDqHq,α,βγgζHq,α,βγgζ−1ζ∈D.Consider(31)ψζ=ζDqHq,α,βγgζHq,α,βγgζζ∈D.The condition (30) may be rewritten into the form
(32)Rψζ−η>kψζ−1ζ∈D.It follows that the range of the expressionψζ, ζ∈D, is a conical domain
(33)Λk,η=w∈ℂ:Rw−η>kw−1,or
(34)Λk,η=w=u+iv∈ℂ:u−η>ku−12+v2,where 0≤k<∞ and 0≤η<1. Note that Λk,η is such that 1∈Λk,η and ∂Λk,η is a curve defined by
(35)∂Λk,η=w=u+iv∈ℂ:u−η2=k2u−12+k2v2.Anyw=u+iv in ∂Λk,η is a quadratic equation in two variables u and v that have no uv term; it is well known that it is a symmetrical conic section about the real axis (for more details, see [1]). It follows that the domain Λk,η is bounded by an ellipse for k>1, by a parabola for k=1 and by a hyperbola if 0<k<1.Finally, fork=0, Λk,η is the right half plane Rw>η. From (30), we obtain that g∈Λk,η if and only if, for ζ∈D,
(36)ζDqHq,α,βγgζHq,α,βγgζ∈Λk,ηζ∈D.Making use of the properties of the domainΛk,η and (36), it follows that if g∈Λk,η, then
(37)RζDqHq,α,βγgζHq,α,βγgζ>k+ηk+1ζ∈D,argζDqHq,α,βγgζHq,α,βγgζ≤arctan1−ηk2−η2k>0,π2k=0.Denote byP the class of analytic and normalized Carathéodory functions and by ψk,η∈P, the function such that ψk,η=Λk,η. Following the notation applied by Ma and Minda [38], for 0≤k<∞ and 0≤η<1, let Pψk,η denote the following class of functions:
(38)Pψk,η=ψ∈P:ψD⊂Λk,η=ψ∈P:ψ≺ψk,ηinD.The functions which play the role of extremal functions for the classPψk,η, see [10] (see also [8, 39]) and are defined by
(39)ψk,ηζ=1+1−2ηζ1−ζk=0,1+21−ηπ2log1+ζ1−ζ2k=1,1−η1−k2cosAkilog1+ζ1−ζ−k2−η1−k20<k<1,1−ηk2−1sinπ2κt∫0uζ/tdx1−x21−t2x2+k2−ηk2−1k>1,with Ak=2/πarccosk, uζ=ζ−t/1−tζ0<t<1,ζ∈D, where t is so such that t=coshπκ′t/4κt and κt is Legendre’s complete elliptic integral of the first kind and κ′t the complementary integral of κt.Obviously, ifk=0, then
(40)ψ0,ηζ=1+21−ηζ+21−ηζ2+⋯.Fork=1, we have (see [40, 41])
(41)ψ1,ηζ=1+8π21−ηζ+163π21−ηζ2+⋯.Using the Taylor series in [1, 4], for 0<k<1, we have
(42)ψk,ηζ=1+1−η1−k2∑m=1∞∑l=12m2lAkl2m−12m−lζm.Finally, whenk>1(43)ψk,ηζ=1+π21−η4tk2−1κ2t1+tζ+4κ2tt2+6t+1−π224tκ2t1+tζ2+⋯,so that, denoting
(44)ψk,ηζ=1+L1ζ+L2ζ2+⋯Lj=Ljk,η;j=1,2,⋯,we get
(45)L1=81−ηarccosk2π21−k20≤k<1,8π21−ηk=1,π21−η4tk2−1κ2t1+tk>1.Letgk,ηζ=ζ+B2ζ2+B3ζ3+⋯ be the extremal function in the class STq,α,βγk,η. Then, the relation between the extremal functions in the classes Pψk,η and STq,α,βγk,η is given by
(46)ψk,ηζ=ζDqHq,α,βγgk,ηζHq,α,βγgk,ηζζ∈D.Making use of (24), (30), and (46), we obtain the following coefficient relation for ψk,ηζ:
(47)mq−1qγ;qm−1q;qm−1Γqαm−1+βBm=∑i=1m−1qγ;qi−1q;qi−1Γqαi−1+βBiLm−i,B1=1.In particular, by a direct computation, we have(48)B2=Γqα+βqγqΓqβL1,(49)B3=Γq2α+βq2γqγ+1qΓqβqL2+L12.Sinceα,β,γ>0,0<q<1 and the Ln′s are nonnegative, it follows that the Bn′s are nonnegative.Theorem 3.
Ifg given by (1) belongs to STq,α,βγk,η, then
(50)b2≤B2,b3≤B3.Proof.
Let(51)ψζ=ζDqHq,α,βγgζHq,α,βγgζζ∈D.
Using the relation (24) for ψζ=1+ρ1ζ+ρ2ζ2+⋯, we have
(52)mq−1qγ;qm−1q;qm−1Γqαm−1+βbm=∑i=1m−1qγ;qi−1q;qi−1Γqαi−1+βbiρm−i,b1=1.
Sinceψk,η is univalent in D, the function
(53)pζ=1+ψk,η−1ψζ1−ψk,η−1ψζ=1+c1ζ+c2ζ2+⋯,is analytic in D and Rqζ>0. From
(54)ψζ=ψk,ηpζ−1pζ+1=1+12c1L1ζ+12c2L1+14c12L2−L1ζ2+⋯,we have
(55)b2=Γqα+βqγqΓqβρ1=Γqα+β2qγqΓqβc1L1≤Γqα+βqγqΓqβL1=B2,where we used the inequality cn≤2 and equation (48). From this relation ρ12+ρ2≤L12+L2 (see [4]) and equation (49), we have
(56)b3=Γq2α+βqρ2+ρ12q2γqγ+1qΓqβ≤Γq2α+βqρ2+ρ12+1−qρ12q2γqγ+1qΓqβ≤Γq2α+βqL2+L12+1−qL12q2γqγ+1qΓqβ=Γq2α+βqL2+L12q2γqγ+1qΓqβ=A3.
So, Theorem3 has been proven.Theorem 4.
Ifg given by (1) belongs to STq,α,βγk,η, then
(57)bm≤∏j=0m−2L1+qjqΓqm−1α+βqm−1∏j=0m−2γ+jqΓqβm≥2.Proof.
The result is clearly true form=2. Let m be an integer with m≥2, and assume that the inequality is true for all i≤m−1. Making use of (47), we have
(58)bm=q;qm−1Γqαm−1+βmq−1qγ;qm−1Γqβϱm−1+∑i=2m−1qγ;qi−1Γqβq;qi−1Γqαi−1+βbiϱm−i≤Γqαm−1+βm−2q!qΓqβ∏j=0m−2γ+jqL1+∑i=2m−1∏j=0i−2γ+jqΓqβi−1q!Γqαi−1+βbiL1≤Γqαm−1+βm−2q!qΓqβ∏j=0m−2γ+jqL11+∑i=2m−1∏j=0i−2γ+jqΓqβi−1q!Γqαi−1+β∏j=0i−2L1+qjqΓqi−1α+βqi−1∏j=0i−2γ+jqΓqβ,where we applied the induction hypothesis to bm and the Rogosinski result ϱ2≤L1 (see [42]). Therefore,
(59)bm≤Γqαm−1+βm−2q!qΓqβ∏j=0m−2γ+jqL11+∑i=2m−1∏j=0i−2L1+qjqqi−1i−1q!.
Applying the principle of mathematical induction, we find that(60)1+∑i=2m−1∏j=0i−2L1+qjqqi−1i−1q!=∏j=1m−2L1+qjqqm−2m−2q!,from which the inequality (57) follows.Similarly, we can prove the following.Theorem 5.
Ifg of the form (1) belongs to the class UCVq,α,βγk,η, then
(61)b2≤B22q,b3≤B32q.Theorem 6.
Ifg of the form (1) belongs to the class UCVq,α,βγk,η, then
(62)bm≤∏j=0m−2L1+qjqΓqm−1α+βqm−1mq∏j=0m−2γ+jqΓqβm≥2.Theorem 7.
Letg∈A be given by (1). If the inequality
(63)∑m=2∞k+1mq−k−ηqγ;qm−1Γqβq;qm−1Γqαm−1+βbm<1−η,holds true, then g∈STq,α,βγk,η.Proof.
Making use of the definition (30) it suffices to prove that
(64)kζDqHq,α,βγgζHq,α,βγgζ−1−RζDqHq,α,βγgζHq,α,βγgζ−1<1−η.
Observe that(65)kζDqHq,α,βγgζHq,α,βγgζ−1−RζDqHq,α,βγgζHq,α,βγgζ−1<k+1ζDqHq,α,βγgζHq,α,βγgζ−1=k+1∑m=2∞mq−1qγ;qm−1Γqβ/q;qm−1Γqαm−1+βbmζm−11+∑m=2∞qγ;qm−1Γqβ/q;qm−1Γqαm−1+βbmζm−1≤k+1∑m=2∞mq−1qγ;qm−1Γqβ/q;qm−1Γqαm−1+βbm1−∑m=2∞qγ;qm−1Γqβ/q;qm−1Γqαm−1+βbm.
The last expression is bounded by1−η if inequality (63) holds.Similarly, we can prove the following.Theorem 8.
Letg∈A be given by (1). If the inequality
(66)∑m=2∞mqk+1mq−k−ηqγ;qm−1Γqβq;qm−1Γqαm−1+βbm<1−η,holds true, then g∈UCVq,α,βγk,η.Now, we need the following lemmas.Lemma 9 (see [38]).
Ifφζ=1+ϰ1ζ+ϰ2ζ2+⋯ is a function with positive real part in D and ν is a complex number, then
(67)ϰ2−νϰ12≤2max1,2ν−1.
The result is sharp for the functions given byφζ=1+ζ2/1−ζ2 or φζ=1+ζ/1−ζ.Lemma 10 (see [38]).
Ifφζ=1+ϰ1ζ+ϰ2ζ2+⋯ is an analytic function with a positive real part in D, then
(68)ϰ2−νϰ12≤−4ν+2ifν≤0,2if0≤ν≤1,4ν−2ifν≥1,when υ<0 or ν>1, the equality holds if and only if φζ=1+ζ/1−ζ or one of its rotations. If 0<ν<1, then the equality holds if and only if φζ=1+ζ2/1−ζ2 or one of its rotations. If ν=0, the equality holds if and only if
(69)φζ=1+λ21+ζ1−ζ+1−λ21−ζ1+ζ0≤λ≤1,or one of its rotations. If ν=1, the equality holds if and only if φ is the reciprocal of one of the functions such that equality holds in the case of ν=0.Also, the above upper bound is sharp, and it can be improved as follows when0<ν<1:
(70)ϰ2−νϰ12+νϰ12≤20≤ν≤12,ϰ2−νϰ12+1−νϰ12≤212≤ν≤1.Theorem 11.
Ifg given by (1) belongs to STq,α,βγk,η, then
(71)b3−μb22≤Γq2α+βL1qγqγ+1qΓqβmax1;L2L1+L1q1−γ+1qΓq2α+βμγqΓqβΓq2α+β.Proof.
Iff∈STq,α,βγk,η, we have
(72)ζDqHq,α,βγgζHq,α,βγgζ≺ψk,ηζ,where ψk,qz is given by (39). From the definition of subordination, we have
(73)ζDqHq,α,βγgζHq,α,βγgζ=ψk,ηwζz∈D,where wζ is a Schwarz function with w0=0 and wζ<1. Let hζ be a function with positive real part in D defined by
(74)hz=1+wζ1−wζ=1+ϰ1ζ+ϰ2ζ2+⋯ζ∈D.
This gives(75)wζ=12ϰ1ζ+12ϰ2−ϰ122ζ2+⋯,(76)ψk,ηwζ=1+12ϰ1L1ζ+12ϰ2L1+14ϰ12L2−L1ζ2+⋯.
Using (76) in (73), we obtain
(77)b2=Γqα+β2qγqΓqβϰ1L1,b3=Γq2α+β2qγqγ+1qΓqβϰ2L1+ϰ12L2−L12+ϰ12L122q.
For any complex numberμ, we have
(78)b3−μb22=Γq2α+βL12qγqγ+1qΓqβϰ2−vc12,where
(79)ν=121−L2L1−L1q1−μγ+1qΓq2α+βγqΓqβΓq2α+β.
Our result now follows by an application of Lemma9. This completes the proof of Theorem 11.Example 1.
Takingq⟶1−, α=k=0, and γ=1 in Theorem 11, we obtain the following result:
Ifg given by (1) satisfies the following inequality
(80)Rζg′ζgζ>η,then
(81)b3−μb22≤1−ηmax1;1+21−η1−2μ.Similarly, we can prove the following theorem for the subclassUCVq,α,βγk,η.Theorem 12.
Ifg given by (1) belongs to UCVq,α,βγk,η, then
(82)b3−μb22≤Γq2α+βL1q3qγqγ+1qΓqβmax1;L2L1+L1q1−3qγ+1qΓq2α+βμ2q2γqΓqβΓq2α+β.Example 2.
Takingq⟶1−, α=k=0, and γ=1 in Theorem 12, we get the following result:
Ifg given by (1) satisfies the following inequality
(83)R1+ζg′′ζg′ζ>η,then
(84)b3−μb22≤1−η3max1;1+21−η1−32μ.Theorem 13.
Let(85)σ1=γqΓqβΓq2α+βL12+qL2−L1γ+1qΓq2α+βL12,σ2=γqΓqβΓq2α+βL12+qL2+L1γ+1qΓq2α+βL12,σ3=γqΓqβΓq2α+βL12+qL2γ+1qΓq2α+βL12.
Ifg given by (1) belongs to the class STq,α,βγk,η, then
(86)b3−μb22≤Γq2α+βqγqγ+1qΓqβL2+L12q1−γ+1qΓq2α+βμγqΓqβΓq2α+βμ≤σ1,Γq2α+βL1qγqγ+1qΓqβσ1≤μ≤σ2,Γq2α+βqγqγ+1qΓqβL2+L12q1−3qγ+1qΓq2α+βμγqΓqβΓq2α+βμ≥σ2.
Further, ifσ1≤μ≤σ3, then
(87)b3−μb22+qγqΓqβΓq2α+βγ+1qΓq2α+βL12L1−L2−L12q1−γ+1qΓq2α+βμγqΓqβΓq2α+βb22≤Γq2α+βL1qγqγ+1qΓqβ.
Ifσ3≤μ≤σ2, then
(88)b3−μb22+qγqΓqβΓq2α+βγ+1qΓq2α+βL12L1+L2+L12q1−γ+1qΓq2α+βμγqΓqβΓq2α+βb22≤Γq2α+βL1qγqγ+1qΓqβ.Proof.
Applying Lemma10 to (78) and (79), we can obtain our results asserted by Theorem 13.Similarly, we can prove the following theorem for the classUCVq,α,βγk,η.Theorem 14.
Let(89)σ4=2q2γqΓqβΓq2α+βL12+qL2−L13qγ+1qΓq2α+βL12,σ5=2q2γqΓqβΓq2α+βL12+qL2+L13qγ+1qΓq2α+βL12,σ6=2q2γqΓqβΓq2α+βL12+qL23qγ+1qΓq2α+βL12.
Ifg given by (1) belongs to the class UCVq,α,βγk,η, then
(90)b3−μb22≤Γq2α+βL1q3qγqγ+1qΓqβL2L1+L1q1−3qγ+1qΓq2α+βμ2q2γqΓqβΓq2α+βμ≤σ4,Γq2α+βL1q3qγqγ+1qΓqβσ4≤μ≤σ5,Γq2α+βL1q3qγqγ+1qΓqβL2L1+L1q1−3qγ+1qΓq2α+βμ2q2γqΓqβΓq2α+βμ≥σ5.
Further, ifσ4≤μ≤σ6, then
(91)b3−μb22+q2q2γqΓqβΓq2α+β3qγ+1qΓq2α+βL12L1−L2−L12q1−3qγ+1qΓq2α+βμ2q2γqΓqβΓq2α+βb22≤Γq2α+βL1q3qγqγ+1qΓqβ.
Ifσ6≤μ≤σ5, then
(92)b3−μb22+q2q2γqΓqβΓq2α+β3qγ+1qΓq2α+βL12L1+L2+L12q1−3qγ+1qΓq2α+βμ2q2γqΓqβΓq2α+βb22≤Γq2α+βL1q3qγqγ+1qΓqβ.Remark 15.
Puttingq⟶1− in the above results, we obtain the corresponding results for the classes STα,βγk,η and UCVα,βγk,η.


## 3. Conclusion

By using the concept of the basic (orq-) calculus, we have introduced two subclasses STq,α,βγk,η and UCVq,α,βγk,η of normalized univalent functions which map the open unit disk D=ζ∈ℂ:ζ<1 onto the generalized conic domain. We have obtained a number of important properties including the coefficient estimates, sufficient conditions, and the Fekete-Szegö inequalities for each of these classes. Our results are connected with those in earlier works which are related to the field of geometric function theory.



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*Source: 1688741-2022-06-09.xml*