Erratum: Certain properties of a subclass of harmonic convex functions of complex order defined by multiplier transformations-Malaya J. Mat. 4(3)2016, 362-372

Abstract

In the paper entitled Certain properties of a subclass of harmonic convex functions of complex order defined by Multiplier transformations- Malaya J. Mat. 4(3)2016, 362-372, the presentation of definition of modified Multiplier transformation of harmonic function \(f=h+\bar{g}\) as given below.
$$
\begin{aligned}
I_\gamma^0 f(z)=D^0 f(z)=h(z)+\overline{g(z)} \\
I_\gamma^1 f(z)=\frac{\gamma D^0 f(z)+D^1 f(z)}{\gamma+1} \\
I_\gamma^n f(z)=I_\gamma^1\left(I_\gamma^{n-1} f(z)\right),\left(n \in N_0\right) \\
I_\gamma^n f(z)=z+\sum_{k=2}^{\infty}\left(\frac{k+\gamma}{1+\gamma}\right)^n a_k z^k+(-1)^n \sum_{k=1}^{\infty}\left(\frac{k-\gamma}{1+\gamma}\right)^n b_k z^k
\end{aligned}
$$
Also if \(f\) is given, then,
\begin{align*}
&I_\gamma^n f(z)=f \widetilde{*} \underbrace{\left(\phi_1(z)+\overline{\phi_2(z)}\right) \widetilde{\ldots \ldots *}\left(\phi_1(z)+\overline{\left.\phi_2(z)\right)}\right.}_{n-\text { times }}\\&=h * \underbrace{\left(\phi _ { 1 } ( z ) * \ldots \left(\phi_1(z)\right.\right.}_{n-\text { times }}+\overline{g+\underbrace{\left(\phi_2(z) * \ldots\left(\phi_2(z)\right)\right)}_{n-\text { times }}},
\end{align*}
where \(*\) denotes the usual Hadamard product or convolution of power series and
$$
\phi_1(z)=\frac{(1+\gamma) z-\gamma z^2}{(1+\gamma)(1-z)^2}, \phi_2(z)=\frac{(\gamma-1) z-\gamma z^2}{(1+\gamma)(1-z)^2}
$$
is taken from the article by Yasar and S. Yalçin [1].