Inequalities involving

  • Horst Alzer

    Waldbröl, Germany
  • Man Kam Kwong

    The Hong Kong Polytechnic University, Hong Kong
  • József Sándor

    Babes-Bolyai University, Cluj-Napoca, Romania
Inequalities involving $\pi(x)$ cover

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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

Cite this article

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

DOI 10.4171/RSMUP/98