Certain convolution operators for meromorphic functions (Q5943074)

The author considers the class \(\Sigma_p\) of Laurent series NEWLINE\[NEWLINEf(z)= z^{-p}+ \sum^\infty_{n=1} a_{n-p} z^{n-p}NEWLINE\]NEWLINE analytic in the unit disk punctured in the origin. Using the usual Hadamard product or convolution and the special function NEWLINE\[NEWLINE\varphi_p (a,c,z)= \sum^\infty_{n=0} {(a)_n\over (c)_n} z^{n-p}, \quad c\neq 0,-1,-2,\dotsNEWLINE\]NEWLINE he defines the linear operator \(L_p (a,c)\) acting on \(\Sigma_p\) by NEWLINE\[NEWLINEL_p(a,c) \bigl(f(z) \bigr)= \varphi_p (a,c,z) *f(z).NEWLINE\]NEWLINE Among other inequalities of similar type a sharp lower bound for \(\text{Re} (z^pL_p (a+1,c) (f(z)))\) is derived for those \(f\in\Sigma_p\) which satisfy NEWLINE\[NEWLINE\text{Re} \biggl((1- \alpha)z^p L_p(a,c) \bigl(f(z) \bigr)+ \alpha z^pL(a+1,c) \bigl(f(z) \bigr)\biggr) >\betaNEWLINE\]NEWLINE for fixed \(\beta<1\), \(\alpha >0\), \(a>0\) and all \(z\) in the unit disk.