Schwarz Lemma and Schwarz-Pick Lemma for solutions of the $\alpha$-harmonic equation

In this paper, the Schwarz type and Schwarz-Pick type inequalities for solutions of $\alpha$-harmonic equation for $\alpha >-1$ are investigated. By making use of the integral of trigonometric functions, we obtain the two types of inequalities in terms of hypergeometric functions which improve the corresponding results due to Khalfallah et al. (Complex Var. Elliptic Equ., 2023) and Li et al. (Bull. Malays. Math. Sci. Soc., 2022).