In this article, we introduce a general family of analytic and bi-univalent functions in the open unit disk, which is defined by applying the principle of differential subordination between analytic functions and the Tremblay fractional derivative operator. The upper bounds for the general coefficients $|a_n|$ of functions in this subclass are found by using the Faber polynomial expansion. We have thereby generalized and improved some of the previously published results.

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...

In this work, the bounds for the logarithmic coefficients ${\gamma }_n$ of the general classes ${\mathcal{S}}^{*}\left(\phi \right)$ and $\mathcal{K}\left(\phi \right)$ were estimated. It is worthwhile mentioning that the given bounds would generalize some of the previous papers. Some consequences of the main results are also presented, noting that our method is more general than those used by others.

The error function takes place in a wide range in the fields of mathematics, mathematical physics and natural sciences. The aim of the current paper is to investigate certain properties such as univalence and close-to-convexity of normalized imaginary error function, which its region is symmetric with respect to the real axis. Some other outcomes are also obtained.

In this paper, we use the Faber polynomial expansion to find upper bounds for |an| (n ≥ 3) coefficients of functions belong to classes HqΣ(λ,h),STqΣ(α,h) andMqΣ(α,h) $\begin{array}{} H_{q}^{\Sigma}(\lambda,h),\, ST_{q}^{\Sigma}(\alpha,h)\,\text{ and} \,\,M_{q}^{\Sigma}(\alpha,h) \end{array}$ which are defined by quasi-subordinations in the open unit disk 𝕌. Further, we generalize some of the previously published results.