Let $\begin{array}{} \mathcal {S}^*_\psi \end{array}$ be a subclass of starlike functions in the unit disk 𝕌, where ψ is a convex function such that ψ(0) = 1, ψ′(0) > 0, ℜ(ψ(ξ)) > 0 and ψ(𝕌) is symmetric with respect to the real axis. We obtain the sharp solution of Fekete-Szegö problem for the family $\begin{array}{} \mathcal {S}^*_\psi \end{array}$, and then extend the result to the case of corresponding subclass defined on the bounded starlike circular domain Ω in several complex variables, which give an unified answer of Fekete-Szegö problem for the kinds of subclasses of starlike mappings defined on Ω. At last, we propose two conjectures related the same problems on the unit ball in a complex Banach space and on the unit polydisk in ℂn.

Abstract:We provide for the first time, a complete list of forbidden minors (obstructions) for the family of graphs with vertex cover 6. This study shows how to limit both the search space of graphs and improve the efficiency of an obstruction checking algorithm when restricted to k-VERTEX COVER graph families. In particular, our upper bounds 2k þ 1 (2k þ 2) on the maximum number of vertices for connected (disconnected) obstructions are shown to be sharp for all k > 0.

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Motivated by the works of H.M. Srivastava et al. [7], we introduce and investigate two new general subclasses HT (?,?,?), h,pHT (?) of bi-starlike and bi-convex of Ma-Minda type functions. Bounds on the first two coefficients |a2| and |a3| for functions in HT (?,?,?), h,pHT (?) are given. The results here generalize and improve the corresponding earlier works done by Ali et al.[1] and Brannan et al.[2].

Abstract. In this paper, we further investigate the class of functions φ * (n, p, λ, α) which are analytic in the open unit disk △ = {z : |z| < 1}, and involve the combinations of the representations of p-valently starlike and convex functions. We obtain several generalized results on the modified-Hadamard product of the class φ * (n, p, λ, α), which extend the corresponding results obtained by Altintaş et al.