engleski

Uniform treatment of Jensen type inequalities

The main purpose of this paper is to give the uniform treatment of the Jensen type inequalities. Using the Green function, we give some new conditions for Jensen's inequality to hold. Integral and discrete case are presented. In the discrete version we get the generalization of the results from [A. McD. Mercer: An "error term" for the Ky Fan inequality, J. Math. Anal. Appl. 220 (1998), 774-777.] and [A. McD. Mercer: Some new inequalities involving elementary mean values, J. Math. Anal. Appl. 229 (1999), 677-681.], and in integral version the generalization of the results from [J. E. Pečarić, I. Perić, H. M. Srivastava: A family of the Cauchy type mean-value theorems, J. Math. Anal. Appl. 306 (2005) 730-739.]. Using the same method we also give adequate results for the converses of the Jensen inequality. As a consequence, also the results concerning the Hermite-Hadamard inequalities are presented.

Green function; Jensen inequality; Hermite-Hadamard inequality

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