# Inequalities involving $π(x)$

### Horst Alzer

Waldbröl, Germany### Man Kam Kwong

The Hong Kong Polytechnic University, Hong Kong### József Sándor

Babes-Bolyai University, Cluj-Napoca, Romania

## Abstract

We present several inequalities involving the prime-counting function $π(x)$. Here, we give two examples of our results. We show that

$916 π(x)π(y)≤π_{2}(x+y)$

is valid for all integers $x,y≥2$. The constant factor $16/9$ is the best possible. The special case $x=y$ leads to

$34 ≤π(x)π(2x) (x=2,3,…),$

where the lower bound $4/3$ is sharp. This complements Landau’s well-known inequality

$π(x)π(2x) ≤2(x=2,3,…).$

Moreover, we prove that the inequality

$(2x+yπ(x+y) )_{s}≤(xπ(x) )_{s}+(yπ(y) )_{s}(0<s∈R)$

holds for all integers $x,y≥2$ if and only if $s≤s_{0}=0.94745….$ Here, $s_{0}$ is the only positive solution of

$(716 )_{t}−(56 )_{t}=1.$

## Cite this article

Horst Alzer, Man Kam Kwong, József Sándor, Inequalities involving $π(x)$. Rend. Sem. Mat. Univ. Padova 147 (2022), pp. 237–251

DOI 10.4171/RSMUP/98