Sciresol Sciresol https://indjst.org/author-guidelines Indian Journal of Science and Technology 0974-5645 10.17485/IJST/v15i43.905 research article On N ew S ubclasses of Analytic Functions Involving (p,q)-Derivatives Shilpa N drshilpamaths@gmail.com 1 Assistant Professor, PG Department of Mathematics, JSS College of Arts Commerce and Science Mysuru, Karnataka, ­570025 India 18 11 2022 15 43 2336 23 5 2021 11 2 2022 2022 Abstract

Objective: The objectives of the present study are to introduce some new subclasses of analytic functions involving (p,q)-derivatives by using subordination. We derive Fekete-Szegӧ inequalities for the functions belonging to the new subclasses. Method: Using the concept of (p,q)-derivative of a function and the subordination principle between analytic functions we introduce and study new subclasses. Findings: The Fekete-Szegӧ problem may be considered as one of the most important results about univalent functions. It was introduced by Fekete-Szegӧ in 1933. Coefficient estimates for the second and third coefficients of functions belonging to class of analytic functions with specific geometric properties were obtained. We obtain the Fekete-Szegӧ inequalities for functions belonging to the new subclasses. Moreover, some special cases of the established results are discussed. Novelty: The results of the paper are new and significantly contribute to the existing literature on the topic.

Keywords Analytic functions Subordination q-calculus Fekete-Szegӧ inequalities (p q)-derivative operator None
Introduction

Let A specify the category of analytic functions fz of the form

fz=z+n=2anzn

in the open unit disc   U=z:zC  and   z<1.

The q-calculus is a generalization of the ordinary calculus without using the limit notation. The theory of q-derivative operators are used in describing and solving various problems in applied science such as ordinary fractional calculus, optimal control, q-difference and q-integral equations, as well as Geometric function theory of complex analysis. The first application and usage of the q-calculus was introduced by Jackson 1, 2. After that many researchers have carried out remarkable studies, which play a significant role in the development of Geometric function theory. One may refer the papers 3, 4, 5, 6, 7, 8, 9, 10, 11 on this subject.

Recently there is an extension of q-calculus, denoted by (p,q)-calculus. The applications of (p,q)-calculus play important role in many diverse areas of the Mathematical, Physical and Engineering sciences. Quite a number of mathematicians 12, 13, 14, 15. For the convenience, we provide some basic definitions and concept details of (p,q)-calculus which are used in this paper. The (p,q)-derivative of the function  fz is defined as 16

D(p,q)f(z)=f(pz)-f(qz)(p-q)z,  (z0,  0<q<p1)

From equation (2) it is clear that if  fz and gz are two functions, then

Dp,qfz+gz =  D(p,q)fz+D(p,q)gz.                                        Dp,qcfz =  cDp,qfz.

Note that Dp,qfzf'z as p =1 and q 1-,  where  f'z is the ordinary derivative of the function fz. Further by (2) the (p,q)-derivative of the function hz=zn, is as follows

Dp,qhz =[n](p,q)zn-1

where [n](p,q) denotes the (p,q)-number and is given as:

[n](p,q)=pn-qnp-q,   0<q<p1 .

Note that [n](p,q)n as p =1 and  q 1-, therefore in view of equation (3),  Dp,qhz= h'(z) as

p =1 and  q 1-,  where  h'(z) denotes the ordinary derivative of the function h(z) with respect to z .

The (p, q)-derivative of the function fz, given by equation (1) is defined as

Dp,qfz =1+n=2[n](p,q)anzn-1 0<q<p1

where [n](p,q) is given by (4).

For the analytic functions  fz and gz in U, we say that the function gz is subordinate to f(z) in U 17, and write gzf(z) if there exists a Schwarz function ω z, which is analytic in U, with ω(0)=0 and |ω(z)|<1 such that

gz =fωz,       z U.

Let P denote the class of all functions φz which are analytic and univalent in U and for which φz is convex with  φ0=1 and  R{φ(z)}>0 for all z  U.

Now using the concept of (p,q)-derivative of a function fzϵA  and the subordination principle between analytic functions we introduce new subclasses of A as follows.

Definition 1.1: A function fz  A is said to be in the class  Rp,qφ if it satisfies the following subordination condition

Dp,qfzφz

where φ(z)  P and 0 < q < p ≤1.

Definition 1.2: A function fz  A is said to be in the class  Np,qφ if it satisfies the following subordination condition

1-αfzz+α  Dp,qfzφz

where φz  P and 0   α 1, 0< q < p ≤1.

Note that for p=1, we have  Rp,qφ= Rqφ 18 and  Np,qφ= Nqφ 18 respectively.

Main results

The Fekete-Szegӧ problem 19 is to obtain the coefficient estimates for the second and third coefficients of functions belonging to class of analytic functions with a specific geometric properties. Now we find the Fekete-Szegӧ inequalities for functions belonging to the classes  Rp,qφ and Np,qφ.

The following lemma is necessary to prove our main results.

Lemma 2.1.20 Let  pz=1+n=1cnzn,     (z   U) be a function with positive real part in U and μ is a complex number, then

c2-μ c122max1; 2μ-1.

The result is sharp for the functions given by pz=1+z1-z and pz=1+z21-z2.

Theorem 2.1: Let φz=1+B1z+B2z2+ ϵ P, with B10. If fz given by (1) belongs to the class  Rp,qφ then

a3-μ a22B1p,qmax1,B2B1-p,qμB1p,q2

where μ is a complex number, and 0 < q < p ≤1. The result is sharp.

Proof: If  f(z)ϵ Rp,qφ, then in view of Definition (1.1) there is a Schwarz function ωz in U with   ω0=0 and ωz<1 in U such that

Dp,qfz=φωz.

We define the function

pz=1+ωz1-ωz=1+p1z+p2z2+

Since ω(z)  is a Schwarz function, we have R{p(z)}>0 and p0=1. Let

gz= Dp,qfz=1+d1z+d2z2+

Using equations (11), (12) and (13) we obtain

gz=φpz-1 pz+1

Since

pz-1 pz+1=1 2p1z+p2-p122z2+p3+p134-p1p2z3+

which gives

φpz-1 pz+1=1+1 2B1p1z+12B1p2-p122+14B2p12z2+

Using equations (13) and (14) we obtain

d1=1 2B1p1                                                                                      d2=12B1p2-p122+14B2p12 .

A simple computation gives

Dp,qfz=1+2p,qa2z+3p,qa3z2+

Inequality (13), yields

d1= p,qa2                                                                                                              d2= p,qa3

now comparing the coefficients of z and z2 and simplifying we get

a2=B1p12p,q

and

a3=B12p,qp2-p122+ B2p124p,q

hence

a3-μ a22=B12p,qp2-γp12

where

γ=121-B2B1-p,qμB1p,q2

Hence, by applying Lemma 2.1, the result follows.

Note that taking p=1 in Theorem 2.1 we get the following result derived in 18.

Corollary 2.1: Let φz=1+B1z+B2z2+ ϵ P, with B10. If fz given by (1) belongs to the class  Rqφ and μ is a complex number, then

a3-μ a22B1qmax1,B2B1-qμB1q2

The result is sharp.

Similarly, we can obtain upper bound for the Fekete-Szegӧ inequalities for functions belonging to the class  Np,qφ as follows.

Theorem 2.2: Let φz=1+B1z+B2z2+ ϵ P, with B10. If fz given by (1) belongs to the class  Np,qφ then

a3-μ a22B1[1-α+3p,qα]max1,B2B1-μB1[1-α+3p,qα][1-α+2p,qα]2

where μ is a complex number, and 0 < q < p ≤1. The result is sharp.

Proof: If f(z)ϵ Np,qφ, then in view of Definition (1.1) there is a Schwarz function ωz in U with   ω0=0 and ωz<1 in U such that

1-αfzz+αDp,qfz=φωz

We define the function

pz=1+ωz1-ωz=1+p1z+p2z2+

Since ω  is a Schwarz function, we have R{p(z)}>0 and p0=1. Let

gz=1-αfzz+αDp,qfz=1+d1z+d2z2+

using equations (15), (16) and (17) we obtain

gz=φpz-1 pz+1

Since

pz-1 pz+1=1 2p1z+p2-p122z2+p3+p134-p1p2z3+

which gives

φpz-1 pz+1=1+1 2B1p1z+12B1p2-p122+14B2p12z2+

using equations (17) and (18) we obtain

d1=1 2B1p1                                                                                            d2=12B1p2-p122+14B2p12

A computation gives

(1-α)f(z)z+αD(p,q)(f(z))=1+(1-α)+p,qαa2z+(1-α)+p,qαa3z2+

Inequality (17), yields

d1=(1-α)+p,qαa2 d2=(1-α)+p,qαa3

or equivalently we get

a2=B1p1 2 1-α+2p,qα

and

a3-μa22=B12(1-α)+p,qαp2-γp12

hence

a3-μ a22=B121-α+3p,qαp2-γp12

Where

γ=121-B2B1-μB1(1-α)+p,qα(1-α)+p,qα2

Hence, by applying Lemma 2.1, the result follows.

Note that taking p=1 in Theorem 2.2 we get the following result derived in 18.

Corollary 2.2: Let φz=1+B1z+B2z2+   P, with B10. If fz is given by (1) belongs to the class  Nqφ and μ is a complex number, then

a3-μ a22B1[1-α+3qα]max1,B2B1-μB1[1-α+3qα][1-α+2qα]2.

The result is sharp.

Conclusion

The q-difference calculus or quantum calculus was initiated at the beginning of 19th century, that was initially developed by Jackson. The q-calculus is one of the tool which is used to introduce and investigate many number of subclasses of analytic functions. The quantum calculus has many applications in the fields of special functions and many other areas. Further there is an extension of the q-calculus to postquantum calculus denoted by the (p,q)-calculus. In this paper we introduce and study new subclasses of analytic functions defined by using (p,q)-derivative operator. We derive the Fekete-Szegӧ inequalities for functions belonging to these classes. Moreover, some special cases of the established results are discussed.