We introduce a new class of bi-univalent functions defined in the open unit disc and connected with a $q$-convolution. We find estimates for the general Taylor-Maclaurin coefficients of the functions in this class by using Faber polynomial expansions and we obtain an estimation for the Fekete-Szegö problem for this class.

MSC 2010: 30C45The universally prestarlike functions of order α ≤ 1 in the slit domain Λ = C [1;∞) have been recently introduced by S. Ruscheweyh. This notion generalizes the corresponding one for functions in the unit disk Δ (and other circular domains in C). In this paper, we obtain the coefficient inequalities and the Fekete-Szegö inequality for such functions.

In this paper, we obtain Fekete–Szegö inequalities for a generalized class of analytic functions $f\left(z\right)\in 𝒜$ for which $1+\frac{1}{b}\left(\frac{z{\left(D_{\alpha ,\beta ,\lambda ,\delta }^{n}f\left(z\right)\right)}^{\text{'}}}{D_{\alpha ,\beta ,\lambda ,\delta }^{n}f\left(z\right)}-1\right)$ ($\alpha ,\beta ,\lambda ,\delta \ge 0$; $\beta >\alpha$; $\lambda >\delta$; $b\in {ℂ}^{*}$; $n\in {\mathbb{N}}_0$; $z\in U$) lies in a region starlike with respect to $1$ and is symmetric with respect to the real axis.

The authors obtain the Fekete-Szegő inequality (according to parameters $s$ and $t$ in the region $s^2+st+t^2<3$, $s\ne t$ and $s+t\ne 2$, or in the region $s^2+st+t^2>3,$$s\ne t$ and $s+t\ne 2$) for certain normalized analytic functions $f\left(z\right)$ belonging to $k{\text{-UST}}_{\lambda ,\mu }^n(s,t,\gamma )$ which satisfy the condition $$\Re \left\{\frac{(s-t)z{\left(D_{\lambda ,\mu }^{n}f\left(z\right)\right)}^{\text{'}}}{D_{\lambda ,\mu }^{n}f\left(sz\right)-D_{\lambda ,\mu }^{n}f\left(tz\right)}\right\}>k\left|\frac{(s-t)z{\left(D_{\lambda ,\mu }^{n}f\left(z\right)\right)}^{\text{'}}}{D_{\lambda ,\mu }^{n}f\left(sz\right)-D_{\lambda ,\mu }^{n}f\left(tz\right)}-1\right|+\gamma ,z\in 𝒰.$$ Also certain...

MSC 2010: 30C45

In the present paper, a theorem for the starlikeness and convexity of multivalent functions involving certain inequalities is given. Some interesting consequences of the main result are also mentioned.

The hereditary properties of convexity and starlikeness for conformal mappings do not generalize to univalent harmonic mappings. This failure leads to the notions of fully starlike and fully convex mappings. In this paper, properties of fully starlike mappings of order α and fully convex mappings of order α (0 ≤ α < 1) are studied; in particular, the bounds for the radius of full starlikeness of order α as well as the radius of full convexity of order α are determined for certain families of...

We obtain several fuzzy differential subordinations by using a linear operator ${ℐ}_{m,\gamma }^{n,\alpha }f\left(z\right)=z+\sum _{k=2}^{\infty }{(1+\gamma (k-1))}^n{m}^{\alpha }{(m+k)}^{-\alpha }{a}_k{z}^k$. Using the linear operator ${ℐ}_{m,\gamma }^{n,\alpha },$ we also introduce a class of univalent analytic functions for which we give some properties.