Asymptotic harmonic-mean conjecture for generalized Taylor means

Let $a,b$ be positive real numbers, let $M_{p,m_{1},m_{2}}(a,b)$ denote the means defined in the paper, and set $n=m_{1}+m_{2}+1$. The harmonic mean is $H(a,b)=\tfrac{2ab}{a+b}$. Asymptotic harmonic-mean conjecture.

$$\lim\limits_{n\rightarrow \infty}M_{p,m_{1},m_{2}}(a,b)=H(a,b).$$

This is proposed as an analogue of the established convergence of the Taylor polynomial means $M_p^r$ to the harmonic mean. The asymptotic behavior of these generalized means is presented as an open direction for future research.