Quasi-Mean Value Theorems for Symmetrically Differentiable Functions

Abstract

In this paper, we give a survey of results related the various quasimean value theorems for symmetrically differentiable functions and present some new results. The symmetric derivative of a real function is discussed and its elementary properties are pointed out. Some results leading to the quasi-Lagrange mean value theorem for the symmetrically differentiable functions are presented along with some generalizations. We also present several results concerning the quasi-Flett mean value theorem for the symmetrically differentiable functions. A new result that eliminates the boundary condition in the quasi-Flett mean theorem is also included. The quasi-Flett mean value theorem of Cauchy like is surveyed along with some related results. A new result that eliminates the boundary condition is presented related to the quasi-Flett mean value theorem of Cauchy like for the symmetrically differentiable functions. Further, by identifying several other new auxiliary functions, we present corresponding new quasi-mean value theorems which are variant of quasi-Lagrange mean value theorem, quasi-Flett mean value theorem, and quasi-Flett mean value theorem of Cauchy like for the symmetrically differentiable functions.