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Some inequalities associated with a linear operator defined for a class of multivalent functions. (English) Zbl 1054.30013
Using the notation $(\lambda)_k$ (for any $\lambda\ge 0$) defined by $$(\lambda)_k= \lambda(\lambda+ 1)\cdots(\lambda+ k-1)\quad (k= 1,2,\dots)$$ define $$\phi_p(a,c,z)= z^p+ \sum^\infty_{k=1} {(a)_k\over (c)_k} z^{k+ p}\ (\vert z\vert< 1,\ a\in\bbfR,\ c\in\bbfR,\ c\ne 0,-1,-2,\dots).$$ Using $\phi_p$ and the Hadamard convolution $*$ define $$L_p(a, c)f(z):= \phi_p(a, c,z)* f(z),$$ where $$f(z)= z^p+ \sum^\infty_{k= p+n} a_k z^k\tag1$$ is an analytic function in the open unit disc $U$ of the complex plane. We shall denote by $A(p,n;a,c,\alpha)$ the class of all analytic functions $f(z)$ of the form (1) satisfying $$\text{Re}\Biggl[{L_p(a+ 2,c)f\over L_p(a+ 1,c)f}\Biggr]< \alpha\ (z\in U,\ \alpha> 1,\ a\in\bbfR,\ c\in\bbfR,\ c\ne 0,-1,-2,\dots).$$ Let $D^{\delta+ p- 1} f$ denote ${z\over (1- z)^{\delta+ p}}*f(z)$. The authors derive various inequalities involving $L_p(b,c,f)$ and $D^\lambda f$ for functions in $A(p,n;a,c,\alpha)$ for various parameters $b$ and $\lambda$. They also obtain results involving differential subordination between analytic functions in their class of functions.
30C45Special classes of univalent and multivalent functions
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