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**Bounds for the normalised Jensen functional.**
*(English)*
Zbl 1113.26021

This is a paper on bounds for the normalized Jensen functional. Jensen’s inequality for convex functions is one of the best known and extensively used inequalities in various fields. 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, Levinson’s inequality and other results. Let

\[ \mathcal P_n : = \left\{ (p_1, p_2, \dots, p_n) \mid p_j \geq 0 \;(j=1, \dots, n),\;\sum_{j=1}^n p_j = 1\right\}. \]

The author considers the normalized Jensen functional

\[ \mathcal J_n(f, X, P): = \sum_{j=1}^n p_j f(x_j) - f\left(\sum_{j=1}^n p_j x_j\right) \geq 0 , \]

where \(f: C \rightarrow \mathbb R\) is a convex function on the convex set \(C\), \(X= (x_1, \dots, x_n) \in C^n\) and \(P=(p_1, \dots, p_n) \in \mathcal P_n\), and proves that if \(P, Q \in \mathcal P_n\), \(q_j >0 \;(j=1, \dots, n)\), then

\[ \max_{1\leq j\leq n} \{{p_j}/{q_j}\} \mathcal J_n(f, X, Q)\geq \mathcal J_n(f, X, P) \geq \min_{1\leq j\leq n}\{{p_j}/{q_j}\} \mathcal J_n(f, X, Q) \geq 0 . \]

As applications, a generalized arithmetic mean-geometric mean-harmonic mean inequality, the positivity of Kullback-Leibler divergence and a refinement of Shannon’s inequality are obtained.

\[ \mathcal P_n : = \left\{ (p_1, p_2, \dots, p_n) \mid p_j \geq 0 \;(j=1, \dots, n),\;\sum_{j=1}^n p_j = 1\right\}. \]

The author considers the normalized Jensen functional

\[ \mathcal J_n(f, X, P): = \sum_{j=1}^n p_j f(x_j) - f\left(\sum_{j=1}^n p_j x_j\right) \geq 0 , \]

where \(f: C \rightarrow \mathbb R\) is a convex function on the convex set \(C\), \(X= (x_1, \dots, x_n) \in C^n\) and \(P=(p_1, \dots, p_n) \in \mathcal P_n\), and proves that if \(P, Q \in \mathcal P_n\), \(q_j >0 \;(j=1, \dots, n)\), then

\[ \max_{1\leq j\leq n} \{{p_j}/{q_j}\} \mathcal J_n(f, X, Q)\geq \mathcal J_n(f, X, P) \geq \min_{1\leq j\leq n}\{{p_j}/{q_j}\} \mathcal J_n(f, X, Q) \geq 0 . \]

As applications, a generalized arithmetic mean-geometric mean-harmonic mean inequality, the positivity of Kullback-Leibler divergence and a refinement of Shannon’s inequality are obtained.

Reviewer: Chun-Gil Park (Daejeon)

### MSC:

26D15 | Inequalities for sums, series and integrals |

26D10 | Inequalities involving derivatives and differential and integral operators |

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\textit{S. S. Dragomir}, Bull. Aust. Math. Soc. 74, No. 3, 471--478 (2006; Zbl 1113.26021)

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### References:

[1] | Pečarić, Convex functions, partial orderings and statistical applications (1992) · Zbl 0749.26004 |

[2] | Mitrinović, Classical and new inequalities in analysis (1993) |

[3] | Bullen, Handbook of mean and their inequalities (2003) · Zbl 1035.26024 |

[4] | Dragomir, Selected topics on Hermite-Hadamard inequalities and applications (2000) |

[5] | DOI: 10.1002/0471200611 |

[6] | McEliece, The theory of information and coding (1977) |

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