In the present paper, we introduce and investigate a new class of analytic and bi-univalent functions f (z) in the open unit disk U. For this purpose, we make use of a linear combination of the following three functions: f(z)/z, f'(z) and z f''(z) for a function belonging to the normalized univalent function class S. By applying the technique involving the Faber polynomials, we determine estimates for the general Taylor-Maclaurin coefficient of functions belonging to the analytic and bi-univalent function class which we have introduced here. We also demonstrate the not-too-obvious behaviour of the first two Taylor-Maclaurin coefficients of such functions.

In the present research manuscript, we formulate a new generalized structure of the nonlinear Caputo fractional quantum multi-integro-differential equation in which such a multi-order structure of quantum integrals is considered for the first time. In fact, in the light of this type of boundary value problem equipped with the multi-integro-differential setting, one can simply study different cases of the existing usual integro-differential problems in the literature. In this direction, we utilize well-known analytical techniques to derive desired criteria which guarantee the existence of solutions for the proposed multi-order quantum multi-integro-differential problem. Further, some numerical examples are considered to examine our theoretical and analytical findings using the proposed methods.

This paper investigates the third Hankel determinant, denoted $H_3\left(1\right)$, for functions within the subclass $\mathcal{R}{\mathcal{S}}_{\sum }^{\mathrm{*}}\left(\lambda \right)$ of bi-univalent functions associated with crescent-shaped regions ${\phi }_{⦅}\left(z\right)=z+\sqrt{1+z^2}$. The primary aim of this study is to establish upper bounds for $H_3\left(1\right)$. By analyzing functions within this specific geometric context, we derive precise constraints on the determinant, thereby enhancing our understanding of its behavior. Our results and examples provide valuable insights into the properties of bi-univalent functions in crescent-shaped domains and contribute to the broader theory of analytic functions.

The paper introduces a new family of analytic bi-univalent functions that are injective and possess analytic inverses, by employing a q-analogue of the derivative operator. Moreover, the article establishes the upper bounds of the Taylor–Maclaurin coefficients of these functions, which can aid in approximating the accuracy of approximations using a finite number of terms. The upper bounds are obtained by approximating analytic functions using Faber polynomial expansions. These bounds apply to both the initial few coefficients and all coefficients in the series, making them general and early, respectively.

The purpose of this paper is to investigate the valency of planar harmonic mappings of bounded boundary rotation of the open unit disc $\mathbb {D}.$ The paper is motivated by the earlier work of the first two authors [Complex Analysis Oper. Theory 5 (2011), 767–774] and the recent work of T. Hayami [Complex Var. Elliptic Equ. 59 (2014), 1214–1222]. First, the authors give a counterexample showing that both the main result of Hayami, Theorem 2.1, and the related conjecture, Conjecture 4.1, are false. Second, the authors give a valency criterion for planar harmonic mappings of bounded boundary rotation of $\mathbb {D}$ , proving an ameliorated statement of Theorem 2.1 and settling a modified version of Conjecture 4.1.