A converse to the neo-classical inequality with an application to the Mittag-Leffler function
A converse to the neo-classical inequality with an application to the Mittag-Leffler function
Abstract We prove two inequalities for the Mittag-Leffler function, namely that the function $$\log E_\alpha (x^\alpha )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>log</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>α</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is sub-additive for $$0<\alpha <1,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo><</mml:mo> <mml:mi>α</mml:mi> <mml:mo><</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> …