MMN-768
Coefficient inequality for certain classes of analytic functions
Abstract
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