An Application of Legendre Polynomials to Bi-Bazilevic Functions associated with q-Ruscheweyh Operator (European Journal of Pure and Applied Mathematics, 2025)
In this paper, we make use of the concept of fractional $q$-calculus to introduce a novel class of bi-Bazilevic functions involving $q$-Ruscheweyh differential operator that are subordinate to Legendre Polynomials. This study explores the characteristics and behaviors of these functions, offering estimates for the modulus of the initial Taylor series coefficients $a_{2}$ and $a_{3}$ within this specific class and its various subclasses. Additionally, the research delves into the traditional Fekete-Szegö functional problem concerning functions $f$ that are part of our newly defined class and several of its subclasses.
Keywords: Bi-Univalent Functions; Bi-Bazilevic; $q$-Ruscheweyh Differential Operator; Jackson $q$-Derivative Operator; $q$-Gamma Function; Fractional $q$-Calculus Operator; Legendre Polynomials; Coefficient estimates; Fekete-Szegö functional problem; Convolution; Hadamard Product