On Classes of Analytic Functions Associated by a Parametric Linear Operator in a Complex Domain

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1 Proceedings of the Pakistan Academy of Sciences: A. Physical and Computational Sciences 54 (3): (207) Copyright Pakistan Academy of Sciences ISSN: (print), (online) Pakistan Academy of Sciences Research Article On Classes of Analytic Functions Associated by a Parametric Linear Operator in a Complex Domain Rabha W. Ibrahim, Adem Kilicman 2, and Zainab E. Abdulnaby 2,3,* Faculty of Computer Science and Information Technology, University of Malaya, 50603, Malaysia 2 Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, UPM Serdang, Selangor, Malaysia 3 Department of Mathematics, College of Science, Al-Mustansiriyah University, Baghdad, Iraq Abstract: The present paper deals with some geometric classes of analytic functions, such as starlike, convex and bounded turning property in a complex domain. These classes are defined by a new linear operator in the normalized space of analytic functions. The linear operator is introduced by a convolution of Szego function involving parametric coefficients type Laguerre polynomial, with the normalized function. Sufficient conditions on this operator are illustrated to study the geometric properties. Our tool is based on some recent results in this direction. The main strategy for this work is to provide parametric functional inequalities in the open unit disk. Keywords: analytic functions, univalent function, starlike function; convex function, bounded turning function, Laguerre polynomial, convolution (or Hadamard product), Riemann zeta function. INTRODUCTION Special function is a major topic in classical analysis which deals with class of mathematical functions and that may arise in the solution of various classical problems of some branch of mathematics and mathematical physics. In particular, most special functions are considered as a function of a complex variable. An intricate special function can be expressed in terms of simpler function and the simplest way to assess a function is to expand it by a Taylor series. Further, special functions have proven their eligibility in associated with an analytic function via convolution technique (or Hadamard product) to define, prove, represent and extend several types of operators, here we suggest a number of the well-known and recent linear operators defined according to special function (see [-5]). Hohlov [6] derived sufficient conditions that guarantee such mappings for the operator defined by means of the Hadamard product with the Gauss hypergeometric function. Carlson and Shaffer [7] adopted the incomplete beta function to define a linear operator, which has been widely used in the space of analytic and univalent functions in the open unit disk UU (see [8]). After more decades Dzoik and Srivastava [9] investigated a convolution linear operator which has been formulated in terms of the generalized hypergeometric function. Several interesting properties and characteristics of the Dziok Srivastava operator are derived (see [0], [20]). Following, by interesting related work, Srivastava [] determined also, another extension covering the Wright function which is known as Srivastava-Wright operator (see [2]). Recently, in Srivastava and Attiya [3] defined an integral operator in terms of convolution with Hurwitz Lerch Zeta function and studied some differential subordination results associated with the operator. It is worth mentioning that, Srivastava-Attiya operator is a generalized of many operators. Received, August 206; Accepted, September 207 *Corresponding author, address: esazainab@yahoo.com

2 258 Rabha W. Ibrahim et al Ibrahim [4] introduced a linear operator for analytic functions derived by using the basic hypergeometric series, also investigated some applications of differential subordinate on Jack s lemma. The paper is organized as follows:section 2 revisesa new linear operator in the normalized space of analytic functions is defined.the linear operator is introduced by a convolution of Szego function involving parametric coefficients type Laguerre polynomial, with the normalized function. Further, some geometric classes of analytic functions, such as starlike, convex and bounded turning property in a complex domain are investigated in Section 3. These classes are sufficient conditions on this operator are illustrated to study the geometric properties. Our method is based on some recent results in this direction. The main strategy of this work is to provide parametric functional inequalities in the open unit disk. 2. PRELIMINARIES Let AA denote the class of functions of the form ff(zz) = zz + aa kk zz kk = zz + aa 2 zz 2 + aa 3 zz 3, aa = () which are analytic in the open unit disk UU = {zz C: zz < } normalized by ff(0) = ff (0) = 0, and let SS be the subclass of the AA of the univalent functions in UU. Further, a function ff(zz) SS is said to be starlike and convex, if their geometric condition satisfies R zz ff (zz) > 0 and R + zz ff (zz) > 0, ff(zz) ff (zz) respectively, these subclasses of SS are denoted by SS and KK. The convolution (or Hadamard product), between two analytic functions, function of the form () and gg(zz) = zz + kk zz kk in UU, is defined by (ff gg)(zz) = zz + aa kk kk zz kk = (gg ff)(zz). (2) A function of bounded turning B if it satisfies the inequality R {ff (zz)} > 0, zz UU. We begin our current consideration by recalling that a generating function of the associated Laguerre polynomials GG(, ττ, zz) defined by Szego [5]. GG(, ττ, zz) = ( zz) ( ττ zz zz) ee = LL () kk (ττ)zz kk kk=0 ( C \{, 2, } ; R() > ; ττ R; zz < ), zz UU (3) where LL kk () (ττ) is the form of generalized Laguerre polynomials of degree kk on the interval (0, ), is given by kk LL () kk (ττ) kk + = kk ii ( ττ) ii ii! ii=0, kk = 0,, = ττ ττ ττ2 2! + (4)

3 On Classes of Analytic Functions Associated by a Parametric Linear Operator 259 where μμ is the generalized binomial coefficient μμ = Γ( +) ΓΓ(μμ + )ΓΓ( μμ + ). Now, for ff AA, C \{, 2, } and ττ R, let define the function Λ(, ττ, zz) by where Λ(, ττ, zz) = GG(, ττ, zz) LL () (ττ), (zz UU) (5) LL () (ττ) = + We proceed to define a linear operator GG ττ AA AA by by the following Hadamard product (2), we obtain GG ττ ff(zz) = zz + Ω,kk (ττ)aa kk zz kk where + ττ 0, 0 GG ττ ff(zz) = Λ(, ττ, zz) ff(zz), (6) (ff AA; zz UU; ττ R; C \{, 2, } ) (zz UU), (7) Ω,kk (ττ) = LL () kk (ττ) () (), LL LL (ττ) (ττ) 0. (8) Remark 2.: From (7) and (8), we have It is clear that Using the relation (7), we obtain and GG ττ 0 ff (zz) = lim 0 {GG ττ ff (zz)}. (9) GG 0 0 ff(zz) = zz + aa kk zz kk = ff(zz) zz GG ττ + ff (zz) = GG ττ ff (zz) ( )GG ττ + ff (zz), aaaaaa GG 0 αα ff(zz) = zz( zz) αα. GG ττ zz ff (zz) = zz GG ττ ff (zz). (0) Our aim is to study the operator (7) in view of the geometric function theory, by introducing some conditions on Ω,kk (ττ). For this purpose we need, the following results. Theorem 2.2: (Nunokawa [6]). If ff(zz) is analytic function in AA, satisfies ff"(zz) <, (zz UU), then Theorem 2.3: (Mocanu [7]). If ff(zz) is analytic function defined by () and satisfies then ff(zz) belongs to SS. ff (zz) < 20 (zz UU), () 5

4 260 Rabha W. Ibrahim et al 3. RESULTS In this section, we concentrate on some results for coefficient of ff(zz) to be in the class SS and for related classes. Theorem 3.: Let 0 ττ < and R() >. If GG ττ ff(zz) AA such that Then, kk 2. kk2(+) Proof: Our aim to apply Theorem 2.2. It is well-known that, when 0 ττ <, the polynomial LL kk () (ττ) takes its maximality at ττ = 0 for all kk 2 and R() >, such that Consequently, we obtain LL kk (0) = kk Γ( + ). Ω,kk (ττ) kk, R() >. By the definition of GG ττ ff(zz), we have the following assertion: [GG ττ ff(zz)]" kk(kk ) aa kk Ω,kk (ττ) kk 2+ aa kk ( ) + kk ( ) ζζ(), R(), where ζζ() is denoted the Riemann zeta function ζζ(ςς) = Thus, [GG ττ ff(zz)]" <, i.e. GG ττ lim( ) ζζ() =. kk=, kk ςς (see [8]). But Theorem 3.2: Let 0 ττ < and R() >. If GG ττ ff(zz) AA such that ζζ(kk) kk 3+, kk 2 where ζζ(kk) > is denoted the Riemann zeta function, then GG ττ

5 On Classes of Analytic Functions Associated by a Parametric Linear Operator 26 Proof: Since [GG ττ ff(zz)]" kk(kk ) aa kk Ω,kk (ττ) kk 2+ aa kk ζζ(kk) = δδ <, kk where δδ = is Euler's constant. Hence, this indicates that GG ττ Theorem 3.3: Let 0 ττ < and R() >. If GG ττ ff(zz) AA such that ζζ(kk) kk 2+, kk 2, ζζ(kk) >, where ζζ(kk) is denoted the Riemann zeta function, then GG ττ Moreover, GG ττ ff(zz) B. Proof: Since [GG ττ ff(zz)]" kk(kk ) aa kk Ω,kk (ττ) kk 2+ aa kk ζζ(kk) Hence, this refers to GG ττ To prove that GG ττ ff(zz) B it is sufficient to show that [GG ττ ff(zz)] > 0. This completes the proof. =, [GG ττ ff(zz)] = + kk aa kk Ω,kk (ττ)zz kk kk + aa kk ζζ(kk) kk = δδ > 0. Theorem 3.4: Let 0 ττ < and R() >. If GG ττ ff(zz) AA such that ζζ(2kk) 2 2kk kk 2+, kk 2, ζζ(kk) >,

6 262 Rabha W. Ibrahim et al where ζζ(2kk) is refereed the Riemann zeta function, then GG ττ Proof: Since Hence, this refers to GG ττ [GG ττ ff(zz)]" kk(kk ) aa kk Ω,kk (ττ) kk 2+ aa kk ζζ(2kk) = 6 <, 2 2kk Theorem 3.5: Let 0 ττ < and R() >. If GG ττ ff(zz) AA such that ζζ(2kk ) (kk )kk +, kk 2, ζζ(kk) >, where ζζ(2kk) is refereed the Riemann zeta function, then GG ττ Proof: Our aim is to apply Theorem 2.3. Since Consequently, we obtain Thus, GG ττ kk aa kk Ω,kk (ττ) kk aa kk Ω,kk (ττ) kk + aa kk ζζ(2kk). (kk ) [GG ττ ff(zz)] ζζ(2kk) kk kk= = log 2 = < 20 5 = Next, we illustrate some geometric properties of the operator GG ττ ff(zz) to be starlike and convex. For this purpose, we recall that the function belongs to SS (ββ), 0 < ββ <, if it achieves the inequality (see [9]). ff(zz) zz ff (zz) 2 ββ < Moreover, the functionff is in the class κκ(ββ), 0 < ββ < if and only if zz ff SS (ββ). We have the following results: Theorem 3.6: Let GG ττ ff(zz) AA, ττ (0,) and R() >. If ββ 0, and 2 kk + 2 ββ + 2 ββ Then, GG ττ ff(zz) SS (ββ).

7 On Classes of Analytic Functions Associated by a Parametric Linear Operator 263 Proof: By the assumption of the theorem, we have the following conclusion: [GG ττ ff(zz)] zz GG ττ ff(zz) 2 ββ [GG ττ ff(zz)] zzgg ττ ff(zz) + kk aa kk kk + aa kk + kk+ aa kk kk + aa kk This yields that GG ττ ff(zz) SS, 0 < ββ < /2. Theorem 3.7: Let GG ττ ff(zz) AA, ττ (0,) and R() >. If ββ 0, and 2 kk +2 2 ββ + 2 ββ then GG ττ ff(zz) KK(ββ). Proof: By the assumption of the theorem, we have the following assertion: zz zz GG ττ ff(zz) 2 ββ zz zz GG ττ ff(zz) + kk+ aa kk kk +2 aa kk + kk+2 aa kk kk +2 aa kk This implies that GG ττ ff(zz) KK(ββ ), 0 < ββ < /2. 2 ββ Theorem 3.8: Let GG ττ ff(zz) AA, ττ (0,) and R() >. If ββ 0, 5 and 6 then GG ττ ff(zz) SS (ββ). kk + Proof: To show that the operator GG ττ ff(zz) SS (ββ), we conclude

8 264 Rabha W. Ibrahim et al zz zz GG ττ ff(zz) 2 ββ + kk+ aa kk kk + aa kk + kk 0 kk kk= 2 kk 0 kk= kk ζζ(0) 2 ζζ(0) = = 3 5, ζζ(0) = 2 This yields that GG ττ ff(zz) SS (ββ), 0 < ββ < 5/6. Theorem 3.9: Let GG ττ ff(zz) AA, ττ (0,) and R() >. If ββ 0, 5 and 6 Then, GG ττ ff(zz) KK(ββ ). kk +2 Proof: To prove that the operator GG ττ ff(zz) KK(ββ ), we conclude zz zz GG ττ ff(zz) 2 ββ + kk+2 aa kk kk +2 aa kk + kk 0 kk kk= 2 kk 0 kk= kk ζζ(0) 2 ζζ(0) = = 3 5, ζζ(0) = 2 This gives that GG ττ ff(zz) KK(ββ ), 0 < ββ < 5/6. Theorem 3.0: Let GG ττ ff(zz) AA, ττ (0,) and R() >. If ββ 0, kk +3, Then, GG ττ ff(zz) SS (ββ). 2 ππ 2 2 (2+ππ 2 ) (0, 0.05) and

9 On Classes of Analytic Functions Associated by a Parametric Linear Operator 265 Proof: To demonstrate that the operator GG ττ ff(zz) SS (ββ), we conclude This gives that GG ττ ff(zz) SS (ββ), 0 < ββ < zz zz GG ττ ff(zz) 2 ββ + kk+ aa kk kk + aa kk + kk 2 kk kk= 2 kk 2 kk= kk ζζ(2) ππ2, ζζ(2) = 2 ζζ(2) 6 2 ββ Theorem 3.: Let GG ττ ff(zz) AA, ττ (0,) and R() >. If ββ 0, Then, GG ττ ff(zz) KK(ββ ). kk +4, 2 ππ 2 2 (2+ππ 2 ) (0, 0.05) and Proof: To show that the operator GG ττ ff(zz) KK(ββ ), we conclude This gives that GG ττ ff(zz) KK(ββ ), 0 < ββ < zz zz GG ττ ff(zz) 2 ββ + kk+2 aa kk kk +2 aa kk + kk 2 kk kk= 2 kk 2 kk= kk ζζ(2) ππ2, ζζ(2) = 2 ζζ(2) 6 Theorem 3.2: Let GG ττ ff(zz) AA, ττ (0,) and R() >. If ββ 0, and 8 kk +4, Then, GG ττ ff(zz) SS (ββ).

10 266 Rabha W. Ibrahim et al Proof: To show that the operator GG ττ ff(zz) SS (ββ), we conclude This gives that GG ττ ff(zz) SS (ββ), 0 < ββ < /8. zz zz GG ττ ff(zz) 2 ββ + kk+ aa kk kk + aa kk + kk 3 kk kk= 2 kk 3 kk= kk ζζ(3), ζζ(3) =.2 2 ζζ(3) Theorem 3.3: Let GG ττ ff(zz) AA, ττ (0,) and R() >. If ββ 0, and 8 kk +5, Then, GG ττ ff(zz) KK(ββ ). Proof: To show that the operator GG ττ ff(zz) SS (ββ), we conclude zz zz GG ττ ff(zz) 2 ββ + kk+2 aa kk kk +2 aa kk + kk 3 kk kk= 2 kk 3 kk= kk ζζ(3), ζζ(3) =.2 2 ζζ(3) This gives that GG ττ ff(zz) KK(ββ ), 0 < ββ < /8. 4. CONCLUSIONS We defined a new linear operator in terms of the generating function of the Laguerre polynomials. The method is concluded by Hadamard product. Based on this operator, we defined new classes of parametric coefficients in the open unit disk. Different studies have been illustrated, involving the geometric properties of these classes. We conclude that the best upper bound of these classes is determined by the Riemann zeta function.

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