Text this: On <mml:math display="inline"><mml:mrow><mml:mo class="MathClass-open">(</mml:mo><mml:mi>M</mml:mi><mml:mo class="MathClass-punc">,</mml:mo><mml:mi>P</mml:mi><mml:mo class="MathClass-close">)</mml:mo></mml:mrow></mml:math>-functions with some features and new inequalities