Subclasses of \(p\)-valent functions defined by convolution involving the \(q\)-hypergeometric function. (English) Zbl 1291.30056

Aldweby, Huda; Darus, Maslina

Transylv. J. Math. Mech. 5, No. 1, 11-21 (2013).

Summary: Motivated by the familiar \(q\)-hypergeometric functions, we introduce here two general subclasses \(S^{p,r,s}_{\lambda,\mu,\eta} (a_i,b_j,q,\alpha,\beta)\) and \(C^{p,r,s}_{\lambda,\mu,\eta}(a_i,b_j,q,\alpha,\beta)\) of analytic \(p\)-valent functions with negative coefficients in the unit disk. These classes generalize well known classes of starlike and convex functions. For these classes, some results which include the coefficient estimates, the integral means inequalities and \(\nu-\rho\) neighborhood are obtained.

MSC:

30C45

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

Keywords:

starlike function; convex function