Summary: Let \(\psi(z)\) be a fixed analytic and univalent function of the form \(\psi(z)=z+\sum_{k=2}^infty c_k z^k\) and \(H_\psi(c_k,\delta)\) be the subclass consisting of analytic and univalent functions \(f\) of the form \(f(z)=z+\sum_{k=2}^\infty a_k z^k\) which satisfy the inequality \(\sum_{k=2}^\infty c_k| a_k|\leq\delta\). In this paper, we determine the sharp lower bounds for \(\mathrm{Re}\left\{\frac {D^p f(z)}{D^p f_n(z)}\right\}\) and \(\mathrm{Re}\left\{\frac {D^p f_n(z)}{D^p f(z)}\right\}\), where \(f_n(z)=z+\sum_{k=2}^\infty a_k z^k\) is the sequence of partial sums of a function \(f(z)=z+\sum_{k=2}^\infty a_k z^k\) belonging to the class \(H_\psi(c_k,\delta)\) and \(D^p\) stands for the Sălăgean derivative.
We extend the results of B. A. Frasin [Acta Math. Acad. Paedagog. Nyházi. (N. S.) 21, 135–145 (2005; Zbl 1092.30019); Appl. Math. Lett. 21, No. 7, 735–741 (2008; Zbl 1152.30308)], T. Rosy et al. [JIPAM, J. Inequal. Pure Appl. Math. 4, No. 4, Paper No. 64, 8 p. (2003; Zbl 1054.30014)], H. Silverman [J. Math. Anal. Appl. 209, No. 1, 221–227 (1997; Zbl 0894.30010)] and we point out that some conditions on the results of Frasin (Theorem 2 in [Zbl 1092.30019] and Theorem 2.7 in [Zbl 1152.30308]) are incorrect and we correct them.