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Starlike harmonic functions in parabolic region associated with a convolution structure. (English) Zbl 1238.30014
Denote by $\Bbb{H}$ the family of functions $$f=h+\overline{g} \tag1$$ which are harmonic, univalent and orientation preserving in the open unit disc $\Bbb{U}=\{z:|z|<1\}$, and assume that $f$ is normalized by $f(0)=f^{\prime}(0)-1=0$. Thus, for $f=h+\overline{g} \in \Bbb{H}$, the functions $h$ and $g$ analytic $\Bbb{U}$ can be expressed in the following forms: $$\displaystyle{h(z)=z+\sum^{\infty}_{n=2}a_{n}z^{n},\quad g(z)=\sum^{\infty}_{n=1}b_{n}z^{n} \quad (0 \leq b_{1}<1),}$$ and $f$ is then given by $$f(z)=z+\sum^{\infty}_{n=2}a_{n}z^{n}+ \overline{\sum^{\infty}_{n=1}b_{n}z^{n}} \quad (0 \leq b_{1}<1).$$ For functions $f \in \Bbb{H}$ given by (1) and $\Bbb{F} \in \Bbb{H}$ given by $$\Bbb{F}(z)=\Bbb{H}(z)+\overline{\Bbb{G}(z)}= z+\sum^{\infty}_{n=2}\Bbb{A}_{n}z^{n}+\overline{\sum^{\infty}_{n=1}\Bbb{B}_{n}z^{n}},$$ we recall that the Hadamard product (or convolution) of $f$ and $\Bbb{F}$ is defined to be $$(f*F)(z)=z+\sum^{\infty}_{n=2}a_{n}\Bbb{A}_{n}z^{n}+ \overline{\sum^{\infty}_{n=1}b_{n}\Bbb{B}_{n}z^{n}} \quad (z \in \Bbb{U}).$$ For the purpose of this paper, the authors introduce a subclass of $\Bbb{H}$ denoted by $\Bbb{R}_{H}(F; \lambda, \gamma)$ which involves convolution and consist of all functions of the form (1) satisfying the inequality: $$\mathrm{Re} \left \{(1+e^{i \psi}) \frac{z(f(z)*F(z))^{\prime}}{(1-\lambda)z+\lambda(f(z)*F(z))}-e^{i \psi} \right \} \geq \gamma.$$ Also they denote $\Bbb{T}_{\Bbb{H}}(F; \lambda, \gamma)=\Bbb{R}_{H}(F; \lambda, \gamma)\bigcap \Bbb{T}_{\Bbb{H}}$, where $\Bbb{T}_{\Bbb{H}}$ is the subfamily of $\Bbb{H}$ consisting of harmonic functions $f=h+\overline{g}$ of the form $$f(z)=z-\sum^{\infty}_{n=2}a_{n}z^{n}+ \overline{\sum^{\infty}_{n=1}b_{n}z^{n}} \quad (0 \leq b_{1}<1).$$ The authors obtain a sufficient coefficient condition for functions $f$ given by (1) to be in the class $\Bbb{R}_{\Bbb{H}}(F; \lambda, \gamma)$. It is shown that this coefficient condition is necessary also for functions belonging to the class $\Bbb{T}_{\Bbb{H}}(F; \lambda, \gamma)$. Further, distortion results and extreme points for functions in $\Bbb{T}_{\Bbb{H}}(F; \lambda, \gamma)$ are also obtained.
##### MSC:
 30C45 Special classes of univalent and multivalent functions 30C50 Coefficient problems for univalent and multivalent functions