The special case of equal with the concave function gives. Here, equality holds iff. Gradshteyn, I. Tables of Integrals, Series, and Products, 6th ed. Hardy, G. Cambridge, England: Cambridge University Press, pp. Jensen, J. Krantz, S.
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Create a free Team What is Teams? Learn more. When Jensen's inequality is equality Ask Question. Asked 7 years, 10 months ago. The proof of Theorem 6 is complete. Theorem 7. Under the assumptions of Theorem 4 , if and , then the inequalities in 19 hold. Theorem 7 can be proved easily by using the idea of Theorem 6 along with the result of Theorem 4. Corollary 2. Under the assumptions of Theorem 6 , the following inequalities hold:. Integrating 19 over , we obtain Remark 3.
If , then obviously the conditions 6 and 7 are satisfied. So, in this case, the inequalities 19 and 26 become The inequalities 18 and 28 can also be found in Remark 2 described in [ 25 ]. In this section, we give some applications of Theorem 6 for power means and quasiarithmetic means. Definition 2. Let and be two nonnegative - tuples and. Then, the power mean of order is defined by We establish the refinement of inequalities involving power means as follows. Corollary 3.
Let , , , and be the nonnegative - tuples, such that. If the - tuple is decreasing and , such that , then the following inequalities hold: where and.
Note that the function is convex on for or. Therefore, utilizing 19 and substituting , and , we get Taking th root of 33 , we obtain which is equivalent to Since the function is concave on for , utilizing Theorem 6 and choosing , and , we obtain Therefore, taking power of 35 , we get Clearly, 36 is equivalent to The inequalities in 31 are proved. Corollary 4. Assume that , , and are nonnegative - tuples, such that.
If the tuple is decreasing and , such that , then we have the following inequalities: where and. Since the function is convex on for or , utilizing 19 and choosing , and , we get Now, taking th root of 43 , we acquire For , one has Inequalities 45 and 44 imply the required inequality Inequality 42 can be obtained by taking the limit of 41 as. Definition 3. If a function is continuous and strictly monotonic, then the quasiarithmetic mean is defined as. Corollary 5.
Suppose that is continuous and strictly monotonic on. Also, suppose that , , and are the positive - tuples with. If is decreasing and the function is convex on , then. If is decreasing and the function is concave, then the inequality 48 holds in the opposite direction.
Using 19 for , and , we get this implies that which is equivalent to The information theory is the mathematical treatment of the concepts, framework, and rule governing the spread of messages through communication systems. The information theory has been used very extensively in the telecommunication systems, and it has also a great role in the study of linguistic studies such as in the sense of speed of reading, length of words, and frequency of words.
In this section, we give some meaningful applications of our main results in the area of information theory. We will establish bounds for the different divergences and Shannon entropy.
We also establish bounds for various distances, Bhattacharyya coefficient, and triangular discrimination. Definition 4 See [ 26 ]. Let and be two positive - tuples, and let be a convex function. Theorem 8.
Suppose that is a convex function, suppose also that , , and are three positive - tuples, such that If is a decreasing tuple, , then. Note that the function is convex, and the conditions of Theorem 6 are satisfied. Therefore, applying Theorem 6 by choosing , , and , for all , we obtain Definition 5 See [ 27 ]. Assume that is a probability distribution; then, the Shannon entropy is defined by. Corollary 6.
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