In this paper we introduce a new subclass of the bi-univalent functions defined in the open unit disc and connected with a q-analogue derivative. We find estimates for the first two Taylor-Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$ for functions in this subclass, and we obtain an estimation for the Fekete-Szegő problem for this function class.

It is well-known that the logarithmic coefficients play an important role in the development of the theory of univalent functions. If denotes the class of functions analytic and univalent in the open unit disk , then the logarithmic coefficients of the function are defined by . In the current paper, the bounds for the logarithmic coefficients for some well-known classes like for and were estimated. Further, conjectures for the logarithmic coefficients for functions belonging to these classes are stated. For example, it is forecasted that if the function...

The zero-truncated Poisson distribution is an important and appropriate model for many real-world applications. Here, we exploit the zero-truncated Poisson distribution probabilities to construct a new subclass of analytic bi-univalent functions involving Gegenbauer polynomials. For functions in the constructed class, we explore estimates of Taylor–Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$, and next, we solve the Fekete–Szegő functional problem. A number of new interesting results are presented to follow upon specializing the parameters involved in our main results.

Using the operator ${\mathcal{L}}_c^a$ defined by Carlson and Shaffer, we defined a new subclass of analytic functions ${\mathcal{ML}}_c^a(\lambda ;\psi )$ defined by a subordination relation to the shell shaped function $\psi \left(z\right)=z+\sqrt{1+z^2}$ . We determined estimate bounds of the four coefficients of the power series expansions, we gave upper bound for the Fekete–SzegőSzegő functional and for the Hankel determinant of order two for $f\in {\mathcal{ML}}_c^a(\lambda ;\psi )$ .