Inequalities for Convex and Non-Convex Functions

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1 pplied Mathematical Sciences, Vol. 8, 204, no. 9, HIKRI Ltd, Inequalities for Convex and Non-Convex Functions Zlatko Pavić Mechanical Engineering Faculty in Slavonski rod University of Osijek, Trg Ivane rlić Mažuranić Slavonski rod, Croatia Copyright c 204 Zlatko Pavić. This is an open access article distributed under the Creative Commons ttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. bstract The paper discusses functions that are similar to convex functions, which may be convex but not necessarily. We consider the application of such functions to convex combinations with the common center, and sets with the common barycenter. The same functions are used in studying the integral quasi-arithmetic means. Mathematics Subject Classification: 265, 26D5 Keywords: convex combination, barycenter, Jensen s inequality. Introduction In summary form, we present the concept of convexity and affinity using binomial combinations. Let X be a real linear space. Let a, b X be points and let α, β R be coefficients. Their binomial combination αa + βb is convex if α, β 0 and if α + β =. If c = αa + βb, then the point c itself is called the combination center. subset of X is convex if it contains all binomial convex combinations of its points. The convex hull convs of a set S X is the smallest convex set which contains S, and it consists of all binomial convex combinations of points of S. Let C X be a convex set. function f : C R is convex if the inequality f(αa+βb αf(a+βf(b holds for all binomial convex combinations αa+βb of pairs of points a, b C.

2 5924 Zlatko Pavić Using the adjective affine instead of convex, requiring the coefficients condition α + β =, and requiring the equality f(αa + βb = αf(a + βf(b, we get a characterization of the affinity. Implementing mathematical induction, we can prove that all of the above applies to k-membered combinations for any positive integer k. We present the discrete and integral form of the famous Jensen s inequality using convex and measurable sets. In 905, applying mathematical induction to convex combinations, Jensen (see [2] has obtained the following. Discrete form of Jensen s inequality. Let C be a convex set of a real linear space. convex function f : C R satisfies the inequality ( k k f α i x i α i f(x i (. for every convex combination k i= α ix i of points x i C. i= In 906, working on transition to integrals, Jensen (see [3] has stated the another form. Integral form of Jensen s inequality. Let be a measurable set of a space with positive measure µ so that µ( > 0. Let I R be an interval. convex function f : I R satisfies the inequality f ( µ( g(x dµ i= µ( f(g(x dµ (.2 for every integrable function g : R satisfying g( I, and providing that f(g is integrable. Numerous papers have been written on Jensen s inequality, different types and variants can be found in [5], [6], [7] and [8]. 2. Discrete Inequalities This section is mostly a part of [7], and in this paper, it is an introduction to work with integrals. The most complete result is Theorem 2.3 relying on the idea of a convex function graph and its secant line. Through the paper we will use an interval I R, and a bounded closed subinterval [a, b] I with endpoints a < b. Every number x R can be uniquely presented as the binomial affine combination x = b x b a a + x a b, (2. b a which is convex if, and only if, the number x belongs to the interval [a, b]. Let I R be an interval containing [a, b], let f : I R be a function,

3 Inequalities for convex and non-convex functions 5925 and let f{a,b} line : R R be the function of the secant line passing through the points (a, f(a and (b, f(b of the graph of f. pplying the affinity of the function f{a,b} line to the combination in (2., we obtain its equation f{a,b}(x line = b x b a f(a + x a f(b. (2.2 b a The consequence of the representations in equations (2. and (2.2 is the fact that every convex function f : I R satisfies the inequality and the reverse inequality f(x f line {a,b}(x for x [a, b], (2.3 f(x f line {a,b}(x for x I \ (a, b. (2.4 Lemma 2.. Let I R be an interval, let [a, b] I be a bounded closed subinterval, let n i= α ia i be a convex combination of points a i [a, b], and let m j= β jb j be a convex combination of points b j I \ (a, b. If the above convex combinations have the common center α i a i = β j b j, (2.5 then the inequality i= α i f(a i i= j= β j f(b j. (2.6 holds for every function f : I R satisfying equations (2.3-(2.4. Proof. To simplify the computation we use the secant equation f{a,b} line (x = κx + λ. Respecting equations (2.3-(2.4, and applying the affinity of f{a,b} line, we get α i f(a i α i (κa i + λ = κ α i a i + λ i= finishing the proof. = κ i= j= β j b j + λ = j= β j f(b j j= i= β j (κb j + λ j= (2.7 Let us look to the relationship between the functions used above and convex functions. Thus, we present the following equivalence. Corollary 2.2. function f : I R is convex if and only if it satisfies the implication (2.5 (2.6 for every bounded closed subinterval [a, b] I.

4 5926 Zlatko Pavić Proof. The proof of necessity is covered by Lemma 2.. The sufficiency will be proved via Jensen s inequality. We take a convex combination c = β j b j (2.8 j= of different points b j I assuming that m 2. Then the pair of points a = b j < b j2 = b exists so that c [a, b] and all b j I \ (a, b. The above combination admits one or two such pairs. Recognizing the center c as one-membered convex combination c, and using equation (2.0 as the implication premise, we get Jensen s inequality ( m f β j b j = f(c β j f(b j (2.9 j= which guarantees the convexity of f. j= Let us finish the section by extending Lemma 2.. Theorem 2.3. Let I R be an interval, let [a, b] I be a bounded closed subinterval, let n i= α ia i be a convex combination of points a i [a, b], and let m j= β jb j be a convex combination of points b j I \ (a, b. If the above convex combinations have the common center αa + βb, then the double inequality α i f(a i αf(a + βf(b β j f(b j. (2.0 i= holds for every function f : I R satisfying equations (2.3-(2.4. Proof. The series of inequalities in equation (2.7 should be extended with the middle term of the equality κ α i a i + λ = αf(a + βf(b = κ β j b j + λ. (2. i= j= j= Then it yields the double inequality in equation ( Main Results The integral analogy of the concept of convex combination is the concept of barycenter. Let µ be a positive measure on R. Let R be a µ-measurable set with µ( > 0. Given the positive integer k, let = k i= ki be the partition of pairwise disjoint µ-measurable sets ki. Taking points x ki ki we determine the convex combination k µ( ki x k = µ( x ki (3. i=

5 Inequalities for convex and non-convex functions 5927 whose center x k belongs to conv. The limit M(, µ of the sequence (x k k is the µ-barycenter of the set. So, the µ-barycenter point ( k µ( ki M(, µ = lim k µ( x ki = x dµ. (3.2 µ( i= The barycenter of the set is in its convex hull, M(, µ conv. If g : R is a µ-integrable function, we similarly define its µ-integral arithmetic mean by the number M (g, µ = g(x dµ. (3.3 µ( The integral arithmetic mean of the function g is in the convex hull of its image, M (g, µ conv{g(}. Lemma 3.. Let µ be a positive measure on R. Let I R be an interval, let [a, b] I be a bounded closed subinterval, and let [a, b] and I \ (a, b be sets of positive measures. If the sets and have the common barycenter then the inequality µ( x dµ = µ( x dµ, (3.4 f(x dµ f(x dµ (3.5 µ( µ( holds for every integrable function f : I R satisfying equations (2.3-(2.4. Proof. The calculation implemented to finite sums in equation (2.7 can also be applied to integrals. Corollary 3.2. Let µ be a positive measure on R. Let I R be an interval, and let I be an inclusion of a bounded interval and a subset of measures 0 < µ( < µ(. If the sets and have the common barycenter, then the double inequality µ( f(x dµ µ( f(x dµ µ( \ \ f(x dµ (3.6 holds for every integrable function f : I R satisfying equations (2.3-(2.4 on the closure [a, b] of the interval. Proof. The sets [a, b] and \ I \ (a, b have positive measures, and so they fulfill the conditions of Theorem 3.3. Since f(x dµ = x dµ (3.7 µ( µ( \ \

6 5928 Zlatko Pavić by equation (3.4, then it follows that f(x dµ µ( µ( \ \ f(x dµ (3.8 by Lemma 3.. To prove the whole inequality in equation (3.6, it remains to show that the middle term is the convex combination of end terms. Using convex combination coefficients α = µ(/µ( and β = µ( \ /µ(, we obtain the equality µ( f(x dµ = concluding the proof. = µ( α µ( ( f(x dµ + f(x dµ + \ β µ( \ The integral analogy of Theorem 2.3 is as follows. f(x dµ \ f(x dµ (3.9 Theorem 3.3. Let µ be a positive measure on R. Let I R be an interval, let [a, b] I be a bounded closed subinterval, and let [a, b] and I \ (a, b be sets of positive measures. If the sets and have the common barycenter αa + βb, then the double inequality f(x dµ αf(a + βf(b f(x dµ (3.0 µ( µ( holds for every integrable function f : I R satisfying equations (2.3-(2.4. Lemma 3. can be generalized by involving additional functions g and h in the following way. Corollary 3.4. Let µ be a positive measure on R. Let I R be an interval, let [a, b] I be a bounded closed subinterval, let, R be sets of positive measures, and let g : R and h : R be integrable functions such that g( [a, b] and h( I \ (a, b. If the functions g and h have the common integral arithmetic mean µ( then the inequality µ( g(x dµ = µ( h(x dµ, (3. f(g(x dµ f(h(x dµ (3.2 µ( holds for every function f : I R satisfying equations (2.3-(2.4, and providing that f(g and f(h are integrable. The above corollary will be used in applications to integral quasi-arithmetic means. It can be more generally stated by using measurable sets and of a space with positive measure µ so that µ( > 0 and µ( > 0.

7 Inequalities for convex and non-convex functions pplications to Quasi-rithmetic Means Functions investigated in this paper can be included to integral quasi-arithmetic means by applying methods such as those for convex functions. Let µ be a positive measures on R. Let R be a measurable set of positive measure, and let g : R be a function. Let I R be an interval, and let ϕ : I R be a strictly monotone continuous function such that g( I, and the composite function ϕ(g is µ-integrable on the set. The quasi-arithmetic mean of the function g respecting the function ϕ and measures µ can be defined by ( M ϕ (g, µ = ϕ ϕ(g(x dµ. (4. µ( If is the interval, then the above number is in. In that case, the term in parentheses belongs to the interval ϕ(, and therefore the quasi-arithmetic mean M ϕ (g, µ belongs to the interval. In order to apply the convexity, we use strictly monotone continuous functions ϕ, ψ : I R such that ψ is convex with respect to ϕ (usually says ψ is ϕ-convex, that is, the function f = ψ(ϕ is convex on the interval ϕ(i. We have the following application of Corollary 3.4 to quasi-arithmetic means. Theorem 4.. Let µ be a positive measure on R. Let I R be an interval, let [a, b] I be a bounded closed subinterval, let, R be sets of positive measures, and let g : R and h : R be integrable functions such that g( [a, b] and h( I \ (a, b. Let ϕ, ψ : I R be strictly monotone continuous functions providing that ϕ(g, ϕ(h, ψ(g, ψ(h are integrable, and let f = ψ(ϕ. If f satisfies equations (2.3-(2.4 and ψ is increasing, and if the equality is valid, then we have the inequality M ϕ (g, µ = M ϕ (h, µ (4.2 M ψ (g, µ M ψ (h, µ. (4.3 Proof. Taking J = ϕ(i, [c, d] = ϕ([a, b], u = ϕ(g and v = ϕ(h, we have u( [c, d] and v( J \(c, d. We will apply Corollary 3.4 to the functions u : R, v : R and f : J R. Using the equality ϕ ( M ϕ (g, µ = ϕ ( M ϕ (h, µ which follows from equation (4.2, and including the functions u and v, we get µ( u(x dµ = µ( v(x dµ. (4.4 Then it follows that f(u(x dµ f(v(x dµ (4.5 µ( µ(

8 5930 Zlatko Pavić by Corollary 3.4. pplying the increasing function ψ to the above inequality, substituting f(u(x with ψ(g(x, and f(v(x with ψ(h(x, we obtain the inequality in equation (4.3. ll the cases of the above theorem are the following. Corollary 4.2. Let the conditions of Theorem 4. be fulfilled. If either f satisfies equations (2.3-(2.4 and ψ is increasing or f satisfies equations (2.3-(2.4 and ψ is decreasing, and if the equality in equation (4.2 is valid, then the inequality in equation (4.3 holds. If either f satisfies equations (2.3-(2.4 and ψ is decreasing or f satisfies equations (2.3-(2.4 and ψ is increasing, and if the equality in equation (4.2 is valid, then the reverse inequality in equation (4.3 holds. special case of the quasi-arithmetic means in equation (4. are power means depending on real exponents r. Thus, using the functions { x r, r 0 ϕ r (x = (4.6 ln x, r = 0 where x (0,, we get the power means of order r in the form ( (g(x r r dµ, r 0 µ( M r (g, µ = (. (4.7 exp ln g(x dµ, r = 0 µ( To apply Theorem 4. to the power means we use the interval of positive real numbers I = (0,. Corollary 4.3. Let µ be a positive measure on R. Let I = (0,, let [a, b] I be a bounded closed interval, let, R be sets of positive measures, and let g : R and h : R be integrable functions such that g( [a, b] and h( I \ (a, b. If M (g, µ = M (h, µ, (4.8 then M r (g, µ M r (h, µ for r (4.9 and M r (g, µ M r (h, µ for r. (4.0 Proof. The proof follows from Theorem 4. and Corollary 4.2 by applying the well-known convex and concave functions such as ϕ(x = x and ψ(x = x r for r 0, and ψ(x = ln x for r = 0. The basic facts relating to quasi-arithmetic and power means can be found in []. For more details on different forms of quasi-arithmetic and power means, as well as their refinements, see [4].

9 Inequalities for convex and non-convex functions 593 cknowledgements This work has been fully supported by Mechanical Engineering Faculty in Slavonski rod. References [] P. S. ullen, D. S. Mitrinović, and P. M. Vasić, Means and Their Inequalities, Reidel, Dordrecht, NL, 988. [2] J. L. W. V. Jensen, Om konvekse Funktioner og Uligheder mellem Middelværdier, Nyt tidsskrift for matematik.., vol. 6, pages 49-68, 905. [3] J. L. W. V. Jensen, Sur les fonctions convexes et les inégalités entre les valeurs moyennes, cta Mathematica, vol. 30, 75-93, 906. [4] J. Mićić, Z. Pavić, and J. Pečarić, The inequalities for quasiarithmetic means, bstract and pplied nalysis, vol. 202, rticle ID 20345, pages 25, 202. [5] M. S. Moslehian, and M. Kian, Jensen type inequalities for Q-class functions, ulletin of the ustralian Mathematical Society, vol. 85, pages 28-42, 202. [6] C. P. Niculescu, and C. I. Spiridon, New Jensen-type inequalities, arxiv: [math.c], pages 7, 202. [7] Z. Pavić, Inequalities on convex combinations with the common center, dvances in Mathematics: Scientific Journal, vol. 3, pages 9, 204 (in Press. [8] Z. Pavić, and S. Wu, Inequalities for convex functions on simplexes and their cones, bstract and pplied nalysis, vol. 204, rticle ID , pages, 204 (in Press. Received: ugust, 204