A Certain q -Sălăgean Differential Operator and Its Applications to Subclasses of Analytic and Bi-Univalent Functions Involving ( p, q )- Chebyshev Polynomia

: In the present investigation, we make use of the q - analogue of the Sălăgean differential operator and introduce a new subclass of analytic and bi-univalent functions S η , µ ∑ involving the ( P , Q ) -Chebyshev polynomials. Furthermore, we derive coefficient inequalities and obtain the Fekete-Szegö problem for this new function class f ∈ S η , µ ∑ of functions


Introduction and motivation
Let A denote the class of functions of the form analytic in the open unit disk U .Also we let S denote the class of all functions in A which are univalent in U .The well known example in this class is the Koebe function k(z), defined by The Bieberbach conjecture about the coefficient of the univalent functions in the unit disk was formulated by Bieberbach [1] in the year 1916.The conjecture states that for every function f ∈ S given by (1), we have |a n | ≤ n, for every n.Strictly inequality holds for all n unless f is the Koebe function or one of its rotation.For many years, this conjecture remained as a challenge to mathematicians.After the proof of |a 3 | ≤ 3 by Lowner in 1923, Fekete-Szegö [2] surprised the mathematicians with the complicated inequality which holds good for all values 0 ≤ µ ≤ 1.Note that this inequality region was thoroughly investigated by Schaefer and Spencer [3].
For a class functions in A and a real (or more generally complex) number µ, the Fekete-Szegö problem is all about finding the best possible constant C(µ) so that a 3 − µ a 2  2 ≤ C(µ) for every function in A .It is well known that every function f ∈ S has a function f −1 , defined by and ).
In fact, the inverse function f −1 is given by Several authors worked on Chebyshev polynomial expansion to find coefficient estimates for bi-univalent functions defined in the open unit disk.Still it attracts more attention on researches in this field.Most recent studies of Altınkaya and S. Yalçın [4] motivated us to define the new class of Sakaguchi type function subordinate to (P, Q)-Chebyshev polynomials.
Let f and g be analytic in the open unit desk U .The function f is subordinate to g written as f ≺ g in U , if there exist a function w analytic in U with w (0) = 0 and |w(z For any integer n ≥ 2 and 0 < Q < P ≤ 1, the (P, Q)−Chebyshev polynomials of the second kind is defined by the following recurrence relations: with the initial values U 0 (x, s, P, Q) = 1 and U 1 (x, s, P, Q) = (PQ)x and s is a variable.In slight view of this recurrence relation, A list of some special cases of the (P, Q)-Chebyshev polynomials of second kind can be defined see [4].These polynomials defined recursively over the integers share numerous interesting properties and have been extensively studied.They have been also found to be topics of interest in many different areas of pure and applied science.The generating function of the (P, Q)-Chebyshev polynomials of the second kind is as follows: where η Q f (z) = f (Qz) are known as Fibonacci operator introduced and studied by [5].Similarly, the operator η P, Q f (z) = f (PQz) was defined in [6].Now let's review some basic definitions and concepts of the q -calculus, that are helpful in our research.Throughout the paper, we asume that 0 < q < 1 and Definition (See [7,8]) Let q ∈ (0, 1) and define the q-number [λ ] q by Definition (See [7,8]) Let q ∈ (0, 1) and define the q-factorial [n] q ! by Definition (See [7,8]) In terms of q-numbers, the Jackson q-exponential function e ω q is defined by Definition (See [7,8]) The q-difference operator denoted as D q f (z) is defined by and D q f (0) = f ′ (0), where q ∈ (0, 1).It can be easily seen that Definition The q-analogue of Sălăgean differential operator (see [9]) R m q f (z) : A → A for m ∈ N, is formed as follows.
The so-called q-polynomials are a significant and fascinating group of special functions, specifically orthogonal polynomials.They can be found in various disciplines of the natural sciences, such as coding theory, discrete mathematics (graph theory and combinatorics), Eulerian series, elliptic functions, theta functions, continuous fractions, and so on (see [10,11]), and algebras and quantum groups (see [12][13][14]).In Srivastava's recently-published survey-cum-expository review article [15], one can find an introductory overview of some important and potential useful developments concerning the Bessel polynomials and the q-Bessel polynomials, as well as a number of other orthogonal polynomials, orthogonal q-polynomials, hypergeometric polynomials, the q-hypergeometric polynomials, and so on.
In the next theorem, we present the Fekete-Szegö inequality for the family S Theorem For 0 ≤ λ ≤ 1 and x, µ ∈ R, let f ∈ A be in the family S Q) x |y (µ)| , |y (µ)| ≤ 1 B .