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

1 Department of Mathematics, Maanshan Teacher’s College, Maanshan 243000, People’s Republic of China
2 Department of Mathematics, Suqian College, Suqian 223800, People’s Republic of China
3 Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
4 Department of Mathematics, Yangzhou University, Yangzhou 225002, People’s Republic of China

Special Issues: Quantum Calculus and Its Applications in Geometric Function Theory

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 $\mathcal{Q}_n(A,B,0)$.
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