Geometric properties of a certain class of multivalent analytic functions associated with the second-order differential subordination

Abstract

We investigate some geometric properties of the class $\mathcal{Q}_n(A,B,\alpha)$ which is defined by the second-order differential subordination and find the sharp lower bound on $|z|=r<1$ for the following functional: $$\mathrm{Re}\left\{(1-\alpha)z^{1-p}f'(z) +\frac{\alpha}{p-1}z^{2-p}f''(z)\right\}$$ over the class $\mathcal{Q}_n(A,B,0)$.

Keywords

• analytic functions
• univalent functions
• second-order differential subordination
• sharp bound
• schwarz lemma
• carathéodory inequality