Inequalities involving $\pi(x)$

Inequalities involving $π (x)$

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

\frac{16}{9} π (x) π (y) \leq π^{2} (x + y)

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

\frac{4}{3} \leq \frac{π ( 2 x )}{π ( x )} (x = 2, 3, \dots),

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

\frac{π ( 2 x )}{π ( x )} \leq 2 (x = 2, 3, \dots) .

Moreover, we prove that the inequality

(2 \frac{π ( x + y )}{x + y})^{s} \leq (\frac{π ( x )}{x})^{s} + (\frac{π ( y )}{y})^{s} (0 < s \in R)

holds for all integers $x, y \geq 2$ if and only if $s \leq s_{0} = 0.94745 \dots .$ Here, $s_{0}$ is the only positive solution of

(\frac{16}{7})^{t} - (\frac{6}{5})^{t} = 1.

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