Beta-Bazilevič Function

In this paper, we introduce a relatively new class, $\mathcal{\mathcal{B}}_{1}^\beta(\alpha)$, namely the class of Beta-Bazilevi\v{c} function is generated by the function Bazilevi\v{c} $\mathcal{B}_{1}(\alpha)$. We introduce the class in question by constructing the Alpha-Convex function, $\mathcal{M}(\alpha)$, introduced by Miller et al. \cite{15}. Using Lemmas of function with positive real part, we were given a sharp estimate of coefficient problems. The coefficient problems to be solved are the modulus of initial coefficients $f$, the modulus of inverse coefficients $f^{-1}$, the modulus of the Logarithmic coefficients $\log \frac{f(z)}{z}$, the Fekete-Szeg\"{o} problem and the second Hankel determinant problem.