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.

The aim of the present paper is to introduce a new class $\mathcal{G}\left(\alpha ,\delta \right)$ of analytic functions in the open unit disk and to study some properties associated with strong starlikeness and close-to-convexity for the class $\mathcal{G}\left(\alpha ,\delta \right)$ . We also consider sharp bounds of logarithmic coefficients and Fekete-Szegö functionals belonging to the class $\mathcal{G}\left(\alpha ,\delta \right)$ . Moreover, we provide some topics related to the results reported here that are relevant to outcomes presented in earlier research.

Using several applications of the theory of differential subordination we obtain sufficient conditions for usually normalized analytic functions to belong to certain subclasses of close-to-convex functions and close-to-convex functions of order $\alpha$ .

In this paper, we introduce a new comprehensive subclass ${\Sigma }_B(\lambda ,\mu ,\beta )$ of meromorphic bi-univalent functions in the open unit disk $\mathbb{U}$. We also find the upper bounds for the initial Taylor-Maclaurin coefficients $|b_0|$, $|b_1|$ and $|b_2|$ for functions in this comprehensive subclass. Moreover, we obtain estimates for the general coefficients $|b_n|(n\geqq 1)$ for functions in the subclass ${\Sigma }_B(\lambda ,\mu ,\beta )$ by making use of the Faber polynomial expansion method. The results presented in this paper would generalize and improve several recent works on the subject.

The purpose of the present work is to determine a bound for the functional $H_2\left(2\right)=a_2{a}_4-a_3^2$ for functions belonging to the class ${\mathcal{C}}_{\Sigma }$ of bi-close-to-convex functions. The main result presented here provides much improved estimation compared with the previous result by means of different proof methods than those used by others.

The logarithmic coefficients play an important role for different estimates in the theory of univalent functions. Due to the significance of the recent studies about the logarithmic coefficients, the problem of obtaining the sharp bounds for the modulus of these coefficients has received attention. In this research, we obtain sharp bounds of the inequality involving the logarithmic coefficients for the functions of the well-known class $\mathcal{G}$ and investigate a majorization problem for the functions belonging to this family. To prove our main results, we use the Briot–Bouquet differential subordination obtained by J.A. Antonino and S.S. Miller and the result of T.J. Suffridge connected to the Alexander integral. Combining these results, we give sharp inequalities for two types of sums involving the modules of the logarithmical coefficients of the functions of the class $\mathcal{G}$ indicating also the extremal function. In addition, we prove an inequality for the modulus of the derivative of two majorized functions of the class $\mathcal{G}$, followed by an application.