Correspondence: [email protected] College of Science and Technology, North China Electric Power University, Baoding, Hebei Province, 071051, People’s Republic of China Abstract In this paper, we extend Hu Ke’s inequality, which is a sharpness of Hölder’s inequality. Moreover, the obtained results are used to improve Hao Z-C inequality and Beckenbach-type inequality that is due to Wang. M...

In this paper, we prove that for $s\in(0,\infty)$, the harmonic mean of $E_k(s)$ and $E_k(1/s)$ is always less than or equal to $\Gamma(1-k,1)$. Where generalized exponential integral function, $\Gamma(u,s)$ upper incomplete gamma function $k\in \mathbb{N}$.