Let

in the open unit disc

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^{ }^{ }

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^{ }^{16}

From equation (2) it is clear that if

Note that

where

Note that

The (p, q)-derivative of the function

where

For the analytic functions ^{17}, and write

Let P denote the class of all functions

Now using the concept of (p,q)-derivative of a function

where

where

Note that for p=1, we have ^{ }^{ }

The Fekete-Szegӧ problem^{ }

The following lemma is necessary to prove our main results.

The result is sharp for the functions given by

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

Proof: If

We define the function

Since

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

Since

which gives

Using equations (13) and (14) we obtain

A simple computation gives

Inequality (13), yields

now comparing the coefficients of z and z^{2} and simplifying we get

and

hence

where

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}.

The result is sharp.

Similarly, we can obtain upper bound for the Fekete-Szegӧ inequalities for functions belonging to the class

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

Proof: If

We define the function

Since

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

Since

which gives

using equations (17) and (18) we obtain

A computation gives

Inequality (17), yields

or equivalently we get

and

hence

Where

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}.

The result is sharp.

The q-difference calculus or quantum calculus was initiated at the beginning of 19^{th} 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