COEFFICIENT INEQUALITIES AND YAMASHITA’S CONJECTURE FOR SOME CLASSES OF ANALYTIC FUNCTIONS

For any real number${\it\beta}$with${\it\beta}>1$, let${\mathcal{M}}(\,{\it\beta})$(${\mathcal{N}}(\,{\it\beta})$respectively) denote the class of analytic functions$f$in the unit disk$\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$of the form$f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$and satisfying$\text{Re}\,P_{f}<{\it\beta}$($\text{Re}\,Q_{f}<{\it\beta}$respectively) in$\mathbb{D}$, where$P_{f}=zf^{\prime }(z)/f(z)$and$Q_{f}=1+zf^{\prime \prime }(z)/f^{\prime }(z)$. Also, for${\it\beta}>1$, let${\mathcal{M}}{\rm\Sigma}(\,{\it\beta})$(${\mathcal{N}}{\rm\Sigma}(\,{\it\beta})$respectively) denote the class of analytic functions$g$of the form$g(z)=z(1+\sum _{n=1}^{\infty }b_{n}z^{-n})$and satisfying$\text{Re}\,P_{g}<{\it\beta}$($\text{Re}\,Q_{g}<{\it\beta}$respectively) for$z\in {\rm\Delta}=\{z\in \mathbb{C}:1<|z|<\infty \}$. In this paper, we shall determine the coefficient bounds, inverse coefficient bounds, the growth and distortion theorem and the upper bounds for the Fekete–Szegő functional${\rm\Lambda}_{{\it\lambda}}(f)=a_{3}-{\it\lambda}a_{2}^{2}$for functions$f$in the classes${\mathcal{M}}(\,{\it\beta})$and${\mathcal{N}}(\,{\it\beta})$. Further, we shall solve the maximal area problem for functions of the type$z/f(z)$when$f\in {\mathcal{M}}(\,{\it\beta})$, which is Yamashita’s conjecture for the class${\mathcal{M}}(\,{\it\beta})$. We shall obtain the radius of convexity for the class${\mathcal{N}}(\,{\it\beta})$. We shall also determine the coefficient bounds for functions$g$in the classes${\mathcal{M}}{\rm\Sigma}(\,{\it\beta})$and${\mathcal{N}}{\rm\Sigma}(\,{\it\beta})$and the inverse coefficient bounds for functions$g$in the class${\mathcal{M}}{\rm\Sigma}(\,{\it\beta})$. All the results are sharp.