𝔹 ^-1 -Convex Functions

Abstract

<jats:p> In this work, <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathbb{B}^{-1}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi mathvariant="double-struck">B</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -convex functions which are a new class of generalized convex functions are studied. Some theorems about important properties of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathbb{B}^{-1}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi mathvariant="double-struck">B</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -convex functions are proved. Also, with the aim of regarding the relation between convex and <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathbb{B}^{-1}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi mathvariant="double-struck">B</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -convex functions, some examples are given. </jats:p>