×

Some inequalities associated with a linear operator defined for a class of analytic functions. (English) Zbl 1131.30325

Summary: We give a sufficient condition on a linear operator \(L_p(a,c)g(z)\) which can guarantee that, for \(\alpha\) a complex number with \(\mathrm{Re}\,\alpha>0\), \[ \mathrm{Re}\left\{(1-\alpha)\frac{L_p(a,c)f(z)}{L_p(a,c)g(z)}+\alpha \frac{L_p(a+1,c)f(z)}{L_p(a+1,c)g(z)}\right\}>\rho,\quad \rho<1, \] in the unit disk \(E\), implies \[ \mathrm{Re}\left\{\frac{L_p(a,c)f(z)}{L_p(a,c)g(z)}\right\}>\rho'>\rho,\quad z\in E. \] Some interesting applications of this result are also given.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
PDFBibTeX XMLCite
Full Text: EuDML EMIS