• Kyungpook Mathematical Journal
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• v.50 no.1
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• pp.37-47
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• 2010

In this present work, the authors obtain Fekete-Szeg$\ddot{o}$ inequality for certain normalized analytic function f(z) defined on the open unit disk for which $\frac{(1-{\alpha})z(D^m_{{\lambda},{\mu}}f(z))'+{\alpha}z(D^{m+1}_{{\lambda},{\mu}}f(z))'}{(1-{\alpha})D^m_{{\lambda},{\mu}}f(z)+{\alpha}D^{m+1}_{{\lambda},{\mu}}f(z)}$ ${\alpha}{\geq}0$) lies in a region starlike with respect to 1 and is symmetric with respect to the real axis. Also certain applications of the main result for a class of functions defined by Hadamard product (or convolution) are given. As a special case of this result, Fekete-Szeg$\ddot{o}$ inequality for a class of functions defined through fractional derivatives is obtained. The motivation of this paper is to generalize the Fekete-Szeg$\ddot{o}$ inequalities obtained by Srivastava et al., Orhan et al. and Shanmugam et al., by making use of the generalized differential operator $D^m_{{\lambda},{\mu}}$.

• Kyungpook Mathematical Journal
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• v.53 no.1
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• pp.13-23
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• 2013

In this present work, the authors obtain Fekete-Szeg$\ddot{o}$ inequality for certain normalized analytic function $f(z)$ defined on the open unit disk for which $$\frac{{\lambda}{\beta}z^3(L(a,c)f(z))^{{\prime}{\prime}{\prime}}+(2{\lambda}{\beta}+{\lambda}-{\beta})z^2(L(a,c)f(z))^{{\prime}{\prime}}+z(L(a,c)f(z))^{{\prime}}}{{\lambda}{\beta}z^2(L(a,c)f(z))^{{\prime}{\prime}}+({\lambda}-{\beta})z(L(a,c)f(z))^{\prime}+(1-{\lambda}+{\beta})(L(a,c)f(z))}\;(0{\leq}{\beta}{\leq}{\lambda}{\leq}1)$$ lies in a region starlike with respect to 1 and is symmetric with respect to the real axis. Also certain applications of the main result for a class of functions defined by Hadamard product (or convolution) are given. As a special case of this result, Fekete-Szeg$\ddot{o}$ inequality for a class of functions defined through fractional derivatives are obtained.