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1 Introduction The most important inequalities concerning convex functions are the following: Theorem 1.1 (Classical Jensen’s inequality, see [ 3 ]) Let g be an integrable function on a probability space ( X , A , μ ) ${(X,\mathcal{A},\mu)}$ taking values in an interval I ⊂ R ${I\subset\mathbb{R}}$ . Then ∫ X g ⁢ 𝑑 μ ${\int_{X}g\,d\mu}$ lies in I . If f is a convex function on I such that f ∘ g ${f\circ g}$ is integrable, then f ⁢ ( ∫ X g ⁢ 𝑑 μ ) ≤ ∫ X f ∘ g ⁢ 𝑑 μ . $f\Bigg{(}\int_{X}g\,d\mu\Biggr{)}\leq\int_{X}f\circ g\,d\mu.$ Theorem 1

Appendix B. The AM-QM Inequality, and Jensen’s Inequality If x1, x2, · · · , xn are any n real numbers, then x1 + x2 + · · · + xn n ≤ √ x21 + x22 + · · · + x2n n with equality iff x1 = x2 = · · · = xn. PROOF. Squaring the arithmetic mean, we have( x1 + x2 + · · · + xn n )2 = x 2 1 + x22 + · · · + x2n + all possible xixj cross-products with i = j n2 . We canwrite this more compactly (andmore clearly, as well, I think) as ( x1 + x2 + · · · + xn n )2 =   n∑ i=1 xi n   2 = n∑ i=1 x2i + n∑ i=1 n∑ j=1 i =j xixj n2 . Now, since the square of a real number is

1 Introduction Jensen’s inequality is one of the most fundamental and extensively used inequalities in analysis and other fields of Mathematics. It is a source of many classical inequalities including the generalized triangle inequality, the arithmetic mean-geometric mean-harmonic mean inequality, the positivity of relative entropy, Shannon’s inequality, Ky Fan’s inequality, Levinson’s inequality and other results. For classical and contemporary developments related to the Jensen inequality, see [1 , 5 , 8 , 9 , 10] where further references are provided

acceptance of the rational expectations hypothesis as “stylized fact” by reviewing empirical evidence and discussing theoretical research. Considering these concerns, two facets are important with regard to applied macroeconomic modeling in general and to our work in particular: (i) preserving the functional form, i. e. no linearization, and (ii) using forecast data for expectations in an economic model. Nevertheless, when using forecast values in connection with non-linearities, another issue arises: Jensen’s inequality (see Jensen , 1906 ). A concave (convex) function

References [1] CERONE, P.-DRAGOMIR, S. S.: A refinement of the Gr¨uss inequality and applications, Tamkang J. Math. 38 (2007), 37-49. Preprint RGMIA Res. Rep. Coll. 5 (2002), Art. 14. http://rgmia.org/papers/v5n2/RGIApp.pdf. [2] CERONE, P.-DRAGOMIR, S. S.: Some applications of de Bruijn’s inequality for power series, Integral Transforms Spec. Funct. 18 (2007), 387-396. [3] DRAGOMIR, S. S.: Discrete Inequalities of the Cauchy-Bunyakovsky-Schwarz Type, Nova Sci. Publ. Inc., Hauppauge, N.Y., 2004. [4] DRAGOMIR, S. S.-IONESCU, N. M.: Some converse of Jensen’s

.pdf]. [7] Dragomir S. S., Some reverses of the Jensen inequality with applications, Bull. Aust. Math. Soc. 87 (2013), no. 2, 177–194. Preprint RGMIA Res. Rep. Coll. 14 (2011), Art. 72. [http://rgmia.org/papers/v14/v14a72.pdf]. [8] Dragomir S. S., A refinement and a divided difference reverse of Jensen’s inequality with applications, Preprint RGMIA Res. Rep. Coll. 14 (2011), Art. 74. [http://rgmia.org/papers/v14/v14a74.pdf]. [9] Dragomir S. S., Ionescu N. M., Some converse of Jensen’s inequality and applications. Rev. Anal. Numér. Théor. Approx. 23 (1994), no. 1

1 Introduction Let C be a convex subset of the linear space X and f be a convex function on C . If p = ( p 1 , ... p n ) is probability sequence and x = ( x 1 , ... x n ) ∈ C n , then (1) f ( ∑ i = 1 n p i x i ) ≤ ∑ i = 1 n p i f ( x i ) $$f\left( {\sum\limits_{i = 1}^n {{p_i}} {x_i}} \right) \le \sum\limits_{i = 1}^n {{p_i}} f({x_i})$$ is well known in the literature as Jensen’s inequality. The Lebesgue integral version of the Jensen inequality is given below: Theorem 1.1 Let (Ω, Λ, μ ) be a measure space with 0 < μ (Ω) < ∞ and let ϕ : I

, New York, 2012. [26] Dragomir S.S., Some trace inequalities for convex functions of selfadjoint operators in Hilbert spaces . Preprint RGMIA Res. Rep. Coll. 17 (2014), Art. 115. Available at http://rgmia.org/papers/v17/v17a115.pdf . [27] Dragomir S.S., Jensen’s type trace inequalities for convex functions of selfadjoint operators in Hilbert spaces . Preprint RGMIA Res. Rep. Coll. 17 (2014), Art. 116. Available at http://rgmia.org/papers/v17/v17a116.pdf . [28] Dragomir S.S., Ionescu N.M., Some converse of Jensen’s inequality and applications , Rev. Anal

., Pečarić J., New generalizations of Popoviciu-type inequalities via new Green’s functions and Montgomery identity, J. Inequal. Appl., 2017, 2017:108 [6] Horváth L., Khan K. A., Pečarić J., Combinatorial improvements of Jensen’s inequality, Monographs in Inequalities, Element, Zagreb, 2014, 8 [7] Horváth L., Khan K. A., Pečarić J., Cyclic refinements of the discrete and integral form of Jensen’s inequality with applications, Analysis, 2016, 36(4), 253-262, DOI: https://doi.org/10.1515/anly-2015-0022 [8] Brnetić I., Khan K. A., Pečarić J., Refinement of Jensen’s inequality

\in\mathbb{T}} , u , υ ∈ C rd ⁢ ( [ a , b ] 𝕋 , ℝ ) {u,\upsilon\in C_{\rm rd}(\mathbb{[}a,b]_{\mathbb{T}},\mathbb{R})} , p > 1 {p>1} and 1 p + 1 q = 1 {\frac{1}{p}+\frac{1}{q}=1} . The following Jensen’s inequality on time scales is given in [ 41 , Theorem 2.2 ]: (2.3) H ⁢ ( ∫ a b | h ⁢ ( s ) | ⁢ g ⁢ ( s ) ⁢ Δ ⁢ s ∫ a b | h ⁢ ( s ) | ⁢ Δ ⁢ s ) ≤ ∫ a b | h ⁢ ( s ) | ⁢ H ⁢ ( g ⁢ ( s ) ) ⁢ Δ ⁢ s ∫ a b | h ⁢ ( s ) | ⁢ Δ ⁢ s , H\Bigg{(}\frac{\int_{a}^{b}|h(s)|g(s)\Delta s}{\int_{a}^{b}|h(s)|\Delta s}% \Bigg{)}\leq\frac{\int_{a}^{b}|h(s)|H(g(s))\Delta s}{\int_{a}^{b}|h(s)|\Delta s