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The focus of the present work is on the establishment and investigation of the coefficient estimates of two new subclasses of bi-close-to-convex functions and bi-concave functions; these are of an Ozaki type and involve a modified Caputo's fractional operator that is associated with three-leaf functions in the open unit disc. The classes are defined using the notion of subordination based on the previously established fractional integral operators and classes of starlike functions associated with a three-leaf function. For functions in these classes, the Fekete-Szegö inequalities and the initial coefficients, | a 2 | and | a 3 | , are discussed. Several new implications of the findings are also highlighted as corollaries. [ABSTRACT FROM AUTHOR]

This paper introduces two novel subclasses of the function class Σ for bi-univalent functions, leveraging generalized telephone numbers and Binomial series through convolution. The exploration is conducted within the domain of the open unit disk. We delve into the analysis of initial Taylor-Maclaurin coefficients | a 2 | and | a 3 | , deriving insights and findings for functions belonging to these new subclasses. Additionally, Fekete-Szegö inequalities are established for these functions. Furthermore, the study unveils a range of new subclasses of Σ , some of which are special cases, yet have not been previously explored in conjunction with telephone numbers. These subclasses emerge as a result of hybrid-type convolution operators. Concluding from our results, we present several corollaries, which stand as fresh contributions in the domain of involution numbers involving hybrid-type convolution operators. [ABSTRACT FROM AUTHOR]

We introduce and examine two new subclass of bi-univalent function Σ , defined in the open unit disk, based on Sălăgean-type q-difference operators which are subordinate to the involution numbers. We find initial estimates of the Taylor–Maclaurin coefficients | a 2 | and | a 3 | for functions in the new subclass introduced here. We also obtain a Fekete–Szegö inequality for the new function class. Several new consequences of our results are pointed out, which are new and not yet discussed in association with involution numbers. [ABSTRACT FROM AUTHOR]

For the first time, we attempted to define two new sub-classes of bi-univalent functions in the open unit disc of the complex order involving Mathieu-type series, associated with generalized telephone numbers. The initial coefficients of functions in these classes were obtained. Moreover, we also determined the Fekete–Szegö inequalities for function in these and several related corollaries. [ABSTRACT FROM AUTHOR]

In this investigation a new subclass of bi-univalent functions is established that are defined in the open unit disk A= {Z e £: |Z| < 1} and are endowed with the Sälägean type ^-difference operator. Then, Hankel inequalities for the new function class are obtained and several related consequences of the results are also stated. [ABSTRACT FROM AUTHOR]