Complementary to Choi-Davis-Jensen type inequalities in external formulas

The classical Jensen inequality is expressed by internally dividing points and so are non- commutative Jensen inequalities. Considering that it is expressed by externally dividing points, we shall discuss complementary to Choi-Davis-Jensen type inequalities from this point of view. \ For example, looking at the external version of the Choi-Davis-Jensen inequality given in [1], we obtain complementary to this inequality under condition that f is a real valued concave function. Moreover, we observe improved and generalized inequalities by using results without concavity [2]. \ References: \ [1] J. I. Fujii, J. Pečarić and Y. Seo, The Jensen inequality in an external formula, J. Math. Inequal., 4 (2012), 473--480. \ [2] J. Mićić, H.R. Moradi and S. Furuichi, Some complementary inequalities to Jensen's operator inequality, J. Inequal. Appl., 2018:25 (2018), 1--20.