## Neighborhood properties of certain classes of multivalently analytic functions associated with the convolution structure.(English)Zbl 1258.30006

Let $$T_{n}(p)$$ denote the class of functions of the form $\displaystyle{f(z)=z^{p}-\sum^{\infty}_{k=n+p} a_{k}z^{k}},\tag{1}$ $$a_{k}\geq 0$$, $$p,n \in \mathbb{N} := \{1, 2, 3, \dots\}$$, which are analytic and $$p$$-valent in the open unit disk $$\mathbb{U}= \{z \in \mathbb{C}:|z|<1\}$$. A function $$f\in T_{n}(p)$$ is said to be $$p$$-valently starlike of order $$\alpha$$, $$0 \leq \alpha<p$$ ($$f \in T^{*}_{n,p}(\alpha)$$), if it satisfies the inequality $\displaystyle{\mathrm{Re} \left(\frac{zf^{\prime}(z)}{f(z)} \right)>\alpha}$ for $$z \in \mathbb{U}$$. Furthermore, a function $$f \in T_{n}(p)$$ is said to be $$p$$-valently convex of order $$\alpha$$, $$0\leq \alpha<p$$ ($$f\in C_{n,p}(\alpha)$$), if it satisfies the inequality $\displaystyle{\mathrm{Re} \left(1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)} \right)>\alpha}$ for $$z \in \mathbb{U}$$. Denote by $$f*g$$ the Hadamard product (or convolution) of the functions $$f$$ and $$g$$, that is, if $$f$$ is given by (1) and $$g$$ is given by $\displaystyle{g(z)=z^{p}+ \sum^{\infty}_{k=n+p}b_{k}z^{k}}, \tag{2}$ where $$p,n \in \mathbb{N}$$, then $\displaystyle{(f*g)(z):=z^{p}- \sum^{\infty}_{k=n+p} a_{k}b_{k}z^{k}=:(g*f)(z)}.$ Let $$S_{n,p}(g; \lambda,\mu, \alpha)$$ denote the subclass of $$T_{n}(p)$$ consisting of functions $$f$$ which satisy the inequality $\displaystyle{\mathrm{Re} \left(\frac{z(F_{\lambda, \mu}*g)^{\prime}(z)}{(F_{\lambda, \mu}*g)(z)} \right) > \alpha ,}$ where $$0\leq \mu \leq \lambda \leq 1$$, $$0\leq \alpha<p$$, $$z \in\mathbb{U}$$ and $F_{\lambda,\mu}(z)=\lambda \mu z^{2}f^{\prime\prime}(z)+(\lambda-\mu)zf^{\prime}(z)+(1-\lambda+\mu)f(z).$ Let $$R_{n,p}(g;\lambda,\mu,\alpha,m,u)$$ denote the subclass of $$T_{n}(p)$$ consisting of functions $$f$$ which satisfy the following non-homogenous Cauchy-Euler differential equation: $\displaystyle{z^{m} \frac{d^{m}w}{dz^{m}}+C^{1}_{m}(u+m-1)z^{m-1} \frac{d^{m-1}w}{dz^{m-1}}+ \dots + C^{m}_{m} w \prod^{m-1}_{j=0}(u+j) =h(z) \prod^{m-1}_{j=0}(u+j+p)},$ where $$w=f(z)\in T_{n}(p)$$, $$h\in S_{n,p}(g;\lambda,\mu,\alpha)$$, $$m \in \mathbb{N}^{*}$$ and $$u\in(-p, \infty)$$. The main object of this paper is to investigate the various properties and characteristics of functions belonging to the above-defined classes $$S_{n,p}(g;\lambda,\mu,\alpha)$$ and $$R_{n,p}(g;\lambda,\mu,\alpha,m,u)$$. Apart from deriving coefficient bounds and distortion inequalities for each of these function classes, the authors establish several inclusion relationships involving the $$(n,\delta)$$-neighborhoods of functions belonging to the general function classes which are introduced above.
Reviewer: Mugur Acu (Sibiu)

### MSC:

 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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### References:

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