Subclasses of meromorphically \(p\)-valent functions involving a certain linear operator (Q2826344)

The author uses the operator of meromorphically \(p\)-valent functions \(L_p^\lambda (a,c)f(z)\), whereNEWLINENEWLINE\[NEWLINEL^\lambda_p (a,c)f(z) = z^{-p} + \sum ^{\infty}_{n=1}\frac {(c)_n(p+\lambda)_n}{(a)_n n!}a_{n-p}z^{n-p}NEWLINE\]NEWLINE was introduced and studied by \textit{M. K. Aouf} et al. [Math. Slovaca 61, No. 6, 907--920 (2011; Zbl 1289.30046)], to define two classes of meromorphically \(p\)-valent functions \(\sum ^{\lambda}_{p,k}(a,c;h)\) and \(K^{\lambda}_{p,k}(a,c;h)\) as follows: \(f\in \sum ^{\lambda}_{p,k}(a,c;h)\) if it satisfies the subordination condition NEWLINE\[NEWLINE - \frac {z\bigl (L^{\lambda}_{p,k}(a,c)f\bigr)'(z)}{pf^{\lambda}_{p,k}(a,c)(z)}\prec h(z), NEWLINE\]NEWLINE \(f\in K^{\lambda}_{p,k}(a,c;h)\) if there exists a function \(g\in \sum ^{\lambda}_{p,k} (a,c;h)\) such that NEWLINE\[NEWLINE - \frac {z\bigl (L^{\lambda}_{p,k}(a,c)f\bigr)'(z)}{\rho g^{\lambda}_{p,k}(a,c)(z)} \prec h(z), \quad h\in \rho, \quad p,k \in \mathbb {N}, NEWLINE\]NEWLINE where \(\rho \) is the class of convex univalent analytic functions \(h(z)\) in \(U=\{z\:| z| < 1\}\) such that \(h(0) = 1\) and \(\Re h(z)> 0\) for \(z \in U\), and \(f^{\lambda}_{p,k} (a,c)(z)\) is given by NEWLINE\[NEWLINE f^{\lambda}_{p,k}(a,c)(z) =\frac {1}{k} \sum ^{k-1}_{j=0}\varepsilon ^{jp}_{k}\bigl (L^{\lambda}_{p,k}(a,c)f\bigr)\Bigl (\varepsilon ^{j}_{k}z\Bigr)\left (\varepsilon_{k}={\operatorname {exp}} \left (\frac {2\pi i}{k}\right), f \in \sum_{p}\right). NEWLINE\]NEWLINE The author uses the results obtained by \textit{P. Eenigenburg} et al. [Prepr., ``Babes-Bolyai'' Univ., Fac. Math., Res. Semin. 4, 1--13 (1983; Zbl 0525.30015)] and \textit{S. S. Miller} and \textit{P. T. Mocanu} [J. Differ. Equations 67, 199--211 (1987; Zbl 0633.34005)] to obtain inclusion relationships for the classes \(\sum ^{\lambda}_{p,k}(a,c;h)\) and \(K^{\lambda}_{p,k}(a,c;h)\).