Exponentially convex functions generated by Wulbert's inequality and Stolarsky-type means

Abstract Let − ∞ a b ∞ . If f is concave on [ a , b ] and ψ ′ is convex on the interval of integration, then Wulbert proved that 1 δ + − δ − ∫ δ − δ + ψ ( u ) d u ≥ 1 b − a ∫ a b ψ ( f ( x ) ) d x , where δ − = f − 3 ( ‖ f ‖ 2 2 − ( f ) 2 ) 1 / 2 , δ + = f + 3 ( ‖ f ‖ 2 2 − ( f ) 2 ) 1 / 2 , f = 1 b − a ∫ a b f ( x ) d x and ‖ f ‖ p = ( 1 b − a ∫ a b | f ( x ) | p d x ) 1 / p . We define new Cauchy type means using a functional defined via above inequality and give some related results as applications.