Inequalities involving
Rendiconti del Seminario Matematico della Università di Padova, Tome 147 (2022), pp. 237-251.
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We present several inequalities involving the prime-counting function . Here, we give two examples of our results. We show that
is valid for all integers . The constant factor is the best possible. The special case leads to
where the lower bound is sharp. This complements Landau’s well-known inequality
Moreover, we prove that the inequality
holds for all integers if and only if Here, is the only positive solution of
@article{RSMUP_2022__147__237_0, author = {Alzer, Horst and Kwong, Man Kam and S\'andor, J\'ozsef}, title = {Inequalities involving $\pi(x)$}, journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova}, pages = {237--251}, volume = {147}, year = {2022}, doi = {10.4171/rsmup/98}, mrnumber = {4450790}, zbl = {1497.11014}, language = {en}, url = {http://www.numdam.org/articles/10.4171/rsmup/98/} }
TY - JOUR AU - Alzer, Horst AU - Kwong, Man Kam AU - Sándor, József TI - Inequalities involving $\pi(x)$ JO - Rendiconti del Seminario Matematico della Università di Padova PY - 2022 SP - 237 EP - 251 VL - 147 UR - http://www.numdam.org/articles/10.4171/rsmup/98/ DO - 10.4171/rsmup/98 LA - en ID - RSMUP_2022__147__237_0 ER -
%0 Journal Article %A Alzer, Horst %A Kwong, Man Kam %A Sándor, József %T Inequalities involving $\pi(x)$ %J Rendiconti del Seminario Matematico della Università di Padova %D 2022 %P 237-251 %V 147 %U http://www.numdam.org/articles/10.4171/rsmup/98/ %R 10.4171/rsmup/98 %G en %F RSMUP_2022__147__237_0
Alzer, Horst; Kwong, Man Kam; Sándor, József. Inequalities involving $\pi(x)$. Rendiconti del Seminario Matematico della Università di Padova, Tome 147 (2022), pp. 237-251. doi : 10.4171/rsmup/98. http://www.numdam.org/articles/10.4171/rsmup/98/
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