MMN-768

Let $T(s,\lambda,\mu,\beta)$ be the class of normalised fuctions defined in the unit disk $\mathbb(U)$ by $T(s,\lambda,\mu,\beta)$ be the class of normalised fuctions defined in the unit disk $\mathbb{U}$ by $$\mathop{\mbox{\rm Re}}\bigg(\frac{\Theta_{\mu}^{\lambda,s+1}f(z)} {\Theta_{\mu}^{\lambda,s}g(z)}\bigg)>0 \qquad (s\in{\mathbb{N}}_{0}=\mathbb{N}\cup\{0\}, z\in\mathbb{U}),$$ for $(\mu\in\mathbb{N},\lambda,s\in{\mathbb{N}}_{0})$, A. Mohammed and M. Darus has already introduced the ooperator $\Theta_{\mu}^{\lambda,s}$ defined by $$\Theta_{\mu}^{\lambda,s}f(z) =z+\sum_{k=2}^{\infty} \frac{(k+\lambda-1)!(\mu-1)!}{\lambda!(k+\mu-2)!} k^sa_kz^k\qquad (\mu\in\mathbb{N}, \lambda,s\in{\mathbb{N}}_{0})$$ and $g\in R_{\beta}(s,\lambda,\mu,\beta)$ the class of normalised functions defined in the unit disk \mathbb{U} by $$ \bigg|\mathop{\mbox{\rm arg}}\bigg( \frac{\Theta_{\mu}^{\lambda,s+1}f(z)} {\Theta_{\mu}^{\lambda,s}f(z)}<\frac{\pi}{2}\beta \bigg)\bigg|\qquad (0<\beta\leq 1). $$ For $f\inT(s,\lambda,\mu,\beta)$ and given by $f(z)=z+a_2z^2+a_3z^3+\dots$, a a sharp upper bound is obtained for $|a_3-\tau a_2^2|$, when $\tau\geq 1$.

Vol. 17 (2016), No. 1, pp. 29-34

DOI: 10.18514/MMN.2016.768

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