We consider the inverse function $z=g(w)$ of a (normalized) starlike function $w=f(z)$ of order $\alpha$ on the unit disk of the complex plane with $0<\alpha<1.$ Krzy{\. z}, Libera and Z\l otkiewicz obtained sharp estimates of the second and the third coefficients of $g(w)$ in their 1979 paper. Prokhorov and Szynal gave sharp estimates of the fourth coefficient of $g(w)$ as a consequence of the solution to an extremal problem in 1981. We give a straightforward proof of the estimate of the fourth coefficient of $g(w)$ together with explicit forms of the extremal functions.

Previous studies found that corilagin inhibits the growth of many cancer cells. Ovarian cancer is the deadliest gynecological malignancy, but research on the effects of corilagin on ovarian cancer is limited. This experiment aims to explore the impact of corilagin on ovarian cancer A2780 cell apoptosis and its mechanism of action to evaluate the prospects of corilagin in treating ovarian cancer. Corilagin significantly affects the A2780 cell cycle, reducing the proportion of G0/G1 and G2/M phase cells and blocking them in the S phase. Corilagin at low concentrations can cause A2780 cell apoptosis, mitochondrial membrane potential decline, and calcium influx. After the cells were stained with Acridine Orange, autophagosomes were observed under a fluorescence microscope and increased with the increase in concentration. The results of transcriptome sequencing showed that the effect of corilagin could cause differential expression of apoptosis-related genes in A2780 cells and concentrated on the PI3K-AKT pathway. The qPCR results showed that the expression of genes P53 and Bax in cells increased while the expression of BCL-2 decreased. The expressions of apoptotic factors caspase-9, caspase-3, p53-upregulated modulator of apoptosis (PUMA), and cytochrome C increased after the action of corilagin, indicating that corilagin can induce cell apoptosis. Corilagin can effectively inhibit the proliferation of A2780 cells, cause cell cycle arrest, and induce cell apoptosis. Moreover, it can trigger autophagy and has in vitro antitumor effects. It is a promising drug for treating ovarian cancer through the PI3K/P53 pathway.