Family of analytic functions of complex order. (English) Zbl 1120.30302

Frasin, B. A.

Acta Math. Acad. Paedagog. Nyházi. (N.S.) 22, No. 2, 179-191 (2006).

Summary: In this paper, we introduce the class \(\mathcal{Q}_{\mathcal{T}}(\Phi,\Psi;\alpha,b)\) of analytic functions of complex order \(b\) and type \(\alpha(0\leq \alpha <1)\). Coefficient inequalities, distortion theorems, closure theorems, radii of close-to-convexity, starlikeness, convexity and fractional calculus for functions belonging to the class \(\mathcal{Q}_{\mathcal{T}}(\Phi,\Psi ;\alpha,b)\) are obtained. Furthermore, we obtain the integral means inequality for the function \(f(z)\) belongs to the class \(\mathcal{Q}_{\mathcal{T}}(\Phi,\Psi ;\alpha,b)\) with the extremal function of this class. Also, we consider \(q\)-\(\delta\)-neighborhood for functions in this class.

MSC:

30C45

Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)

Keywords:

analytic functions; coefficient inequalities; distortion theorems; fractional calculus; integral means; neighborhoods