ON APPROXIMATE HERMITE HADAMARD TYPE INEQUALITIES

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1 ON APPROXIMATE HERMITE HADAMARD TYPE INEQUALITIES JUDIT MAKÓ AND ATTILA HÁZY Abstract. The main results of this paper offer sufficient conditions in order that an approximate lower Hermite Hadamard type inequality implies an approximate Jensen convexity property. The key for the proof of the main result is a Korovkin type theorem. 1. Introduction Throughout this paper R, R +, N and Z denote the sets of real, nonnegative real, natural and integer numbers respectively. Let X be a real linear space and D X be a convex set. Denote by D the set D D. One can easily see that, for any constant ε, the ε-convexity of f cf. [7], i.e., the validity of f tx + 1 ty tfx + 1 tfy + ε x, y D, t [, 1], implies the following lower and upper ε-hermite Hadamard inequalities x + y 1 1 f f tx + 1 ty dt + ε x, y D, and 1 f tx + 1 ty dt fx + fy + ε x, y D. The above implication was discovered if ε = by Hadamard [5] in See also [16], [9] and [19] for a historical account. For ε =, the converse is also known to be true cf. [18], [19], i.e., if a function f : D R which is continuous over the segments of D satisfies 1 or with ε =, then it is also convex. Concerning the reversed implication for the case ε >, Nikodem, Riedel, and Sahoo in [] have recently shown that the ε-hermite Hadamard inequalities 1 and do not imply the cε-convexity of f with any c >. Thus, in order to obtain results that establish implications between the approximate Hermite Hadamard inequalities and the approximate Jensen inequality, one has to consider these inequalities with nonconstant error terms. Let α J : D R be an even error function. We say that a function f : D R is α J -Jensen convex, if for all x, y D, x + y 3 f fx + fy + α J x y x, y D. 1 Mathematics Subject Classification. Primary 39B, 39B1, 6A51, 6B5. Key words and phrases. convexity, approximate convexity, lower and upper Hermite Hadamard inequalities. The first named author research was supported by the European Union and the State of Hungary, co-financed by the European Social Fund in the framework of TÁMOP A/ National Excellence Program. The second named author research has been supported by the János Bolyai research scholarship of the Hungarian Academy of Sciences. 1

2 J. MAKÓ AND A. HÁZY In order to describe the old and new results about the connection of an approximate Jensen convexity inequality and the approximate Hermite Hadamard inequality with variable error terms, we need to introduce the following terminology. For a function f : D R, we say that f is hemi-p, if, for all x, y D, the mapping 4 t f1 tx + ty t [, 1] has property P. For example f is hemiintegrable, if for all x, y D the mapping defined by 4 is integrable. Analogously, we say that a function h : D R is radially-p, if for all u D, the mapping t htu t [, 1] has property P on [, 1]. In [1] the authors established the connections between an upper Hermite Hadamard type inequality and a Jensen type inequality, which were stated in the following theorem. Theorem A. Let α H : D R be even and radially upper semicontinuous, ρ : [, 1] R + be integrable with 1 ρ = 1 and there exist c and p > such that ρt c ln 1 t p 1 t ], 1 [ ] 1, 1[, and λ [, 1]. Then every f : D R lower hemicontinuous function satisfying the approximate upper Hermite Hadamard inequality 1 f tx + 1 ty ρtdt λfx + 1 λfy + α H x y fulfills the approximate Jensen inequality 3, provided that α J semicontinuous solution of the functional inequality x, y D, : D R is a radially lower and α J α H. α J u 1 α J 1 t uρtdt + α H u u D In [6], A. Házy and Zs. Páles established a connection between a lower Hermite Hadamard type inequality and a Jensen type inequality by proving the following result. Theorem B. Let α H : D R + be a nonnegative even function. Assume that f : D R is an upper hemicontinuous function satisfying the approximate lower Hermite Hadamard inequality x + y 1 5 f f tx + 1 ty dt + α H x y x, y D, Then f is α J -Jensen convex, where α J : D R + is a nonnegative radially lower semicontinuous, radially increasing solution of the functional inequality 6 α J u 1 α J tudt + α H u u D. In [13] using a Korokvkin type theorem the authors prove the following theorem.

3 ON APPROXIMATE HERMITE HADAMARD TYPE INEQUALITY 3 Theorem C. Let µ be a Borel probability measure on [, 1] with a non-singleton support. Let ε : D R such that εx, x = for all x D and ε : D [, 1] R be a function such that, for all x, y D, ε x, y, = ε x, y, 1 = and ε x, y, s ε x, y, st µ 1 dµt + ε x, µ 1 s µ 1 x + s µ 1 y s [, µ 1 ], ε x, y, t+s st µ 1 1 µ 1 dµt + ε 1 s 1 µ 1 x + s µ 1 1 µ 1 y, y s [µ 1, 1]. Then every f : D R upper hemi-continuous solution of the following lower Hermite Hadamard type functional inequality fµ 1 x + 1 µ 1 y ftx + 1 tydµt + εx, y x, y D also fulfills ftx + 1 ty tfx + 1 tfy + ε x, y, t x, y D, t [, 1]. In this paper we examine the implication from an upper Hermite Hadamard type inequality to a Jensen type inequality. Thus in this paper, we are searching connections between the approximate upper Hermite Hadamard inequality 7 ftx + 1 tydµt λfx + 1 λfy + α H x y and the approximate Jensen inequality 3, where f : D R, α H, α J : D R are given even functions, λ R and µ is a Borel probability measure on [, 1]. First we prove a Korovkin-type theorem Theorem 1, Proposition 4, then in Theorem 5 below, we generalize Theorem A replacing the Lebesgue Stieltjes integral by an integral with respect to an arbitrary Borel probability measure. Throughout this paper, the notation δ t stands for the Dirac measure concentrated at the point t [, 1].. Korovkin type theorems In the sequel, denote by C[, 1] and B[, 1] the space of continuous and bounded Borel measurable real valued functions defined on the interval [, 1] equipped with the usual supremum norm. Denote by p i : [, 1] R the following polynomials: p i u := u i, i =, 1, Let µ be a Borel probability measure on [, 1] and denote by µ 1 the first momentum of µ, namely tdµt. In this section, we will prove a Korovkin type theorem, which will play an important role in the proof of the main result Theorem 5. To see the historical background of these theorems, we recall the classical Korovkin theorem [8], [1], which play an important role in the functional analysis. Theorem D. Let T n : B[, 1] B[, 1] be a sequence of positive operators such that for all u [a, b] and i {, 1, } lim T np i u = p i u.

4 4 J. MAKÓ AND A. HÁZY Then, for all bounded upper semicontinuous function h : [, 1] R 8 lim sup T n hu hu u [a, b]. A simple example of the sequence of this operator is the classical Bernstein-operators, namely n n k 9 B n hu := h u k n k 1 u n k u [, 1]. It is easy to see that k= B n p = p, B n p 1 = p 1 and B n p = 1 n p 1 + n 1 n p. The following theorem is the main application of Theorem D, this is Weierstrass s first approximation theorem, which says that for all continuous function on a compact interval can be approximated by polynomials: Theorem E. Using the above notations, for all h C[, 1], lim B n h = h. In [13] Zs. Páles and the first author got the following Korovkin type theorem. Theorem F. Let T n : B[, 1] B[, 1] be a sequence of positive operators such that for all u [a, b] and i {, 1} lim T np i u = p i u. Suppose that there exists a strictly convex g C[, 1], such that T n g gp + g1 gp 1 Then, for all bounded upper semicontinuous function h : [, 1] R 1 lim supt n h hp + h1 hp 1. In [13], an important example has also been given, namely let h tu µ 1 dµt if u µ1, 11 T µ hu := h 1 1 t1 u 1 µ 1 dµt if µ1 u 1. and T n := T n µ. It can be proved that the sequence of these operators has the property 1. Using these facts Theorem C can be also proved. See [13] for more details. The first main result of this paper the following Korovkin type result. Theorem 1. Let T n : B[, 1] B[, 1] n N be a sequence of positive linear operators such that 1 lim T n p = p. Suppose that there exists a function g C[, 1] with g 1 = and g > on [, 1] \ { 1 } such that lim T n g = p. Then, for all bounded lower semicontinuous function h : [, 1] R, 13 lim inf T nh h 1 p.

5 ON APPROXIMATE HERMITE HADAMARD TYPE INEQUALITY 5 Remark. It can be easy to see under the same conditions of the previous one, if h : [, 1] R is upper semicontinuous, then we get: 14 lim sup T n h h 1 p. It easily follows from the above theorem that, if f is continuous, then 13 holds with equality and the liminf can be replaced by lim. Proof. Let h B[, 1] be a lower semicontinuous function and ε > be arbitrary. φu := ε + hu h 1, u [, 1] \ { 1 gu }. Since h is lower semicontinous continuous in 1, there exists < δ < 1 Thus ε < hu h 1 if u 1 < δ. φ > on ]δ 1, 1 [ ] 1, δ + 1 [. such that: The function φ is lower semicontinuous and the set [, δ 1] [δ + 1, 1] is compact, therefore there exists a K R, such that: φ > K on [, δ 1 ] [δ + 1, 1]. Thus, there exists L such that φ > L on [, 1] \ { 1 }. Therefore, ε Lgu < hu h 1 u [, 1]. Using that T n is a positive linear functional on B[, 1] we can get that Taking the limit n, we get εt n p LT n g < T n h h 1 T np. ε lim inf T nh h 1 T np. Since ε > is arbitrary, this implies that which means that 13 holds. lim inf T nh h 1 T np. In what follows, we construct a large family of positive linear operators on B[a, b] which satisfies the assumptions of the previous results and will be instrumental in the investigation of approximate convexity. Let µ be a Borel probability measure on [, 1] and define a sequence of linear operators T n µ : B[a, b] B[a, b] by the following formula: 15 T nh µ :=... h 1 + 1t t n 1 dµt 1... dµt n p. Proposition 3. Assume that µ is a Borel probability measure on [, 1] and define T µ n by 15. Then, for all n N, T µ n : B[a, b] B[a, b] is a bounded positive linear operator with 16 T µ n 1. In addition, for all h B[, 1] 17 T µ np = p.

6 6 J. MAKÓ AND A. HÁZY Proof. If h B[a, b] then, for all fixed u [, 1] and n N, T nhu µ... h 1 + 1t t n 1 dµt1... dµt n p u h... p udµt 1... dµt n p u = h p, which proves the boundedness of T µ n and 16. The linearity and positivity of T µ n is obvious. Proposition 4. Assume that µ is a Borel probability measure on [, 1], such that µ / {αδ + 1 αδ 1 α [, 1]} and for all n N define T µ n by 15. Then, for all lower semicontinuous h B[, 1], 18 h 1 lim inf Tµ nhu u [, 1]. Proof. By Proposition 3 we can get that T µ n : B[, 1] B[, 1] is a positive linear operator which satisfies 1. Let g : [, 1] R be defined by gt := t 1 t [, 1]. Then, g 1 = and g is positive on [, 1] \ { 1 }. On the other hand, T nh µ = 1... t t n 1 dµt1... dµt n p = t 1 dµt n p. Since h 1, it is easy to see that t 1 dµt 1. To prove that this inequality is strict, assume that t 1 dµt = 1. Then t 1 = 1 µ-almost every where t [, 1], which means that t = or t = 1 µ-almost every where t [, 1]. This implies that µ{, 1} = 1, which is a contradiction by assumptions. Thus, t 1 dµt < 1 and we can get that lim T µ nh =. Therefore, by Theorem 1 we get Hermite Hadamard type inequalities The next theorem gives a connection between an approximate upper Hermite Hadamard type inequality and a Jensen type inequality. Theorem 5. Assume that µ is a Borel probability measure on [, 1], such that µ / {αδ + 1 αδ 1 α [, 1]}. Let λ R and α H : D R be an even error function and assume and f : D R is a hemi-bounded, lower hemicontinuous and, for all x, y D, satisfies the following Hermite Hadamard type inequality 7. Then f is approximate Jensen-convex in the sense of 3, where α J : D R is a radially-bounded and radially upper semicontinuous solution the following functional inequality: 19 α H u + α J 1 t udµt α J u u D, providing that α J α H.

7 ON APPROXIMATE HERMITE HADAMARD TYPE INEQUALITY 7 The proof of Theorem 5 is similar as the proof of the main theorem of [1] and it is based on a sequence of the following lemmata. Lemma 6. Let α H : D R be even, µ is a Borel probability measure on [, 1], such that µ / {αδ + 1 αδ 1 α [, 1]} and λ R. Then every f : D R lower hemicontinuous function satisfying the approximate Hermite Hadamard inequality 3, fulfills f tx + 1 ty + f 1 tx + ty dµt fx + fy + α H x y x, y D. Proof. Changing the role of x and y in 3, then adding the inequality so obtained to the original inequality 3, by the evenness of α H, we get that In what follows, we examine the Hermite Hadamard inequality. For a sequense t n, n N define the following sequense by induction, 1 T 1 := t 1 and T n+1 := 1 t n+1 T n + t n+1 1 T n. Lemma 7. Let T n be definied by 1, then T n = 1 1 t 1 1 t n 1. Proof. We prove by induction on n. If n = 1, T 1 = t 1 and t 1 1 = t 1, so the statement is obvious. Assume that is true for n = k and prove for n = k + 1. By 1 and the inductive assumption we can get that T k+1 = 1 t k+1 T k + t k+1 1 T k 1 = 1 t k+1 + 1t t k 1 + t k t t k = t t k 1 t k+1, which proves this lemma. Lemma 8. Let α H : D R be a radially upper semicontinuous function. If f : D R fulfills the approximate Hermite Hadamard inequality then, for all n N, the function f also satisfies the Hermite Hadamard inequality 1... f T n x + 1 T n y + f 1 T n x + T n y dµt 1... dµt n 3 fx + fy + α n x y for all x, y D, whenever n N, where the sequences T n and α n : D R are defined by 1 and 4 α 1 = α H, α n+1 u = α n 1 t udµt + α H u u D, respectively. Proof. We prove by induction on n. If n = 1, by the definition of α 1, 3 holds. Let x, y D and assume that 3 holds for some n N. Write x by 1 t n+1 x+t n+1 y and y by t n+1 x+1 t n+1 y

8 8 J. MAKÓ AND A. HÁZY in 3. Using the definition of T n+1 and α n+1 1,4 and using also that α H is even, then we obtain: 1... f T n+1 x + 1 T n+1 y + f 1 T n+1 x + T n+1 y dµt 1... dµt n f1 t n+1x + t n+1 y + ft n+1 x + 1 t n+1 y + α n 1 t n+1 x y. Integrating with respect to t n+1, and applying the inductive assumption, and finally 4, we obtain that 1... f T n+1 x + 1 T n+1 y + f 1 T n+1 x + T n+1 y dµt 1... dµt n dµt n+1 1 f1 t n+1 x + t n+1 y + ft n+1 x + 1 t n+1 y dµt n+1 + α n 1 t n+1 x ydµt n+1 fx + fy + α H x y + fx + fy + α n+1 x y, which proves the statement. α n 1 t n+1 x ydµt n+1 Lemma 9. Let α H : D R be even, µ is a Borel probability measure on [, 1], such that µ / {αδ + 1 αδ 1 α [, 1]}. If f : D R is hemibounded and lower hemicontinuous function, then 5 lim inf 1... Proof. Let x, y D be fixed and define h : [, 1] R by f T n x + 1 T n y + f 1 T n x + T n y dµt 1... dµt n x + y f. ht := 1 f 1 tx + ty + f tx + 1 ty. Since f is hemibounded and lower hemicontinuous, h is lower semicontinuous and h B[, 1]. Using also Lemma 7 we have that the operator T n µ defined by 15 can be expressed as, T nh µ =... ht n dµt 1... dµt n. By Proposition 4, 18 holds, which means that 5 that also holds. Lemma 1. Let α H : D R be a radially upper semicontinuous function. Then, for all n N, the function α n : D R defined by 4 is nondecreasing [nonincreasing], whenever α H

9 ON APPROXIMATE HERMITE HADAMARD TYPE INEQUALITY 9 is nonnegative [nonpositive]. Furthermore, if α J : D R is a radially bounded and radially upper semicontinuous solution of the functional inequality 19 then 6 lim sup α n u α J u α J + α H u D. Proof. Assume first that α H is nonnegative. We will prove by induction on n N, that the sequence α n is nondecreasing, i.e., 7 α n+1 α n n N. For n = 1, by the nonnegativity of α 1 = α H, we have that, for all u D, α u = α 1 1 t udµt + α H u α 1 u. Assume that 7 holds for some n N and consider the case n + 1. Using the definition of α n+1, the inductive assumption and the nonnegativity of α n, we get, for all u D, that α n+ u = α n+1 1 t udµt + α H u α n 1 t udµt + α H u = α n+1 u. Analogously, if α H is nonpositive, we can obtain that the sequence α n is nonincreasing. To prove 6, let α J : D R be a radially upper semicontinuous solution of 19. Then, for the sequence of functions g n := α n α J, we obtain g n+1 u g n 1 t udµt u D, n N, Iterating this inequality, similarly as in Lemma 9 and Lemma 8, it can be proved that 8 g n+1 u... g 1 1 T n udµt 1 dµt n u D, n N, where T n is defined by 1. Taking the limsup as n in 8, by Proposition 4, we get that, for all u D, lim sup g n+1 u lim sup... g 1 1 T n udµt 1 dµt n g 1 = α H α J, which immediately yields 6. Proof of Theorem 5. Assume that the conditions of Theorem 5 hold. Using Lemma 8, we obtain 3. Then taking the lim inf in 3 and using the fact that lim inf α n lim sup α n, then applying Lemma 9 and Lemma 1 we obtain that the function f is α J -Jensen convex, i.e. 3 holds. A simple consequence of Theorem 5 is the following corollary which is a generalized form of Theorem A [1]. Corollary 11. Let α H : D R be even and radially bounded and radially upper semicontinuous, ρ : [, 1] R + be integrable with 1 ρ = 1 and λ R. Then every f : D R hemi-bounded

10 1 J. MAKÓ AND A. HÁZY and lower hemicontinuous function satisfying the approximate upper Hermite Hadamard inequality 1 f tx + 1 ty ρtdt λfx + 1 λfy + α H x y x, y D, fulfills the approximate Jensen inequality 3 provided that α J : D R is a radially lower semicontinuous solution of the functional inequality and α J α H. α J u 1 α J 1 t uρtdt + α H u u D In what follows let X be a normed space, ν be a signed Borel measure on ], [ and define the error function by the following way: 9 α H u := u q dνq ], [ These error functions determine a large class of the error functions. Corollary 1. Assume that ν is a signed Borel measure on ], [, such that 3 u q dνq < u D and 31 ], [ ], [ 1 1 t q dµt 1 d ν q <. Let µ be a Borel probability measure on [, 1], such that µ / {αδ + 1 αδ 1 α [, 1]}. Let λ R and assume and f : D R is hemi-bounded and lower hemicontinuous and, for all x, y D, satisfies the following Hermite Hadamard type inequality: 3 ftx + 1 tydµt λfx + 1 λfy + x y q dνq. Then f is approximate Jensen-convex in the following sense: x + y 1 fx + fy f t dµt q x y q dνq x, y D. ], [ Proof. By Theorem 5, it suffices to show that the function 1 α J u := t q dµt u q dνq u D ], [ is well-defined and satisfies 19 with equality where α H : D R is defined by α H u := u q dνq u D. ], [ ], [

11 ON APPROXIMATE HERMITE HADAMARD TYPE INEQUALITY 11 To see that, for all u D, αu is finite, we distinguish two cases. If u 1, then u q 1 for all q >, and hence, by assumption 31, 1 α J u 1 1 t dµt q d ν q <. ], [ Now let u > 1. Then, the functions q u q and q 1 functions, hence α J u u ], [ 1 1 t q dµt are increasing t dµt q d ν q t dµt ]1, [ u q d ν q, which is again finite by conditions 3 and 31. To prove that α J satisfies 19, using that µ is a probability measure, we compute α H 1 s udµs + α J u = t q dµt 1 s u q dνqdµs + u q dνq ], [ = 1 s q dµs 1 1 t q dµt + 1 u q dνq = α J u, ], [ ], [ which proves that 19 holds with equality. References [1] F. Altomare and M. Campiti. Korovkin-type approximation theory and its applications de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 17, MR g:411, [] M. Bessenyei. Hermite Hadamard-type inequalities for generalized convex functions. J. Inequal. Pure Appl. Math., 93:Article 63, pp. 51 electronic, 8. [3] M. Bessenyei and Zs. Páles. Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl., 63:379 39, 3. [4] M. Bessenyei and Zs. Páles. Characterizations of convexity via Hadamard s inequality. Math. Inequal. Appl., 91:53 6, 6. [5] J. Hadamard. Étude sur les propriétés des fonctions entières et en particulier d une fonction considérée par Riemann. J. Math. Pures Appl., 58:171 15, [6] A. Házy and Zs. Páles. On a certain stability of the Hermite Hadamard inequality. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 4651: , 9. [7] D. H. Hyers and S. M. Ulam. Approximately convex functions. Proc. Amer. Math. Soc., 3:81 88, 195. [8] P. P. Korovkin. On convergence of linear positive operators in the space of continuous functions Doklady Akad. Nauk SSSR N.S., 9: , [9] M. Kuczma. An Introduction to the Theory of Functional Equations and Inequalities, volume 489 of Prace Naukowe Uniwersytetu Śl askiego w Katowicach. Państwowe Wydawnictwo Naukowe Uniwersytet Śl aski. Warszawa-Kraków-Katowice, [1] J. Makó and Zs. Páles. Approximate convexity of Takagi type functions. J. Math. Anal. Appl., 369: , 1. [11] J. Makó and Zs. Páles. Approximate Hermite Hadamard type inequalities for approximately convex functions. Math. Inequal. Appl., 16:57-56, 13.

12 1 J. MAKÓ AND A. HÁZY [1] J. Makó and Zs. Páles. Implications between approximate convexity properties and approximate Hermite Hadamard inequalities. Cent. Eur. J. Math., 1: , 1. [13] J. Makó and Zs. Páles. Korovkin type theorems and approximate Hermite Hadamard inequalities. J. Approx. Theory, 8: , 1. [14] J. Makó and Zs. Páles. On approximately convex Takagi type functions. Proc. Amer. Math. Soc., 6:69 8, 13. [15] J. Makó and Zs. Páles. On ϕ-convexity. Publ. Math. Debrecen, 8:17 16, 1. [16] D. S. Mitrinović and I. B. Lacković. Hermite and convexity. Aequationes Math., 8:9 3, [17] C. T. Ng and K. Nikodem. On approximately convex functions. Proc. Amer. Math. Soc., 1181:13 18, [18] C. P. Niculescu and L.-E. Persson. Old and new on the Hermite-Hadamard inequality. Real Anal. Exchange, 9: , 3/4. [19] C. P. Niculescu and L.-E. Persson. Convex Functions and Their Applications. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 3. Springer-Verlag, New York, 6. A contemporary approach. [] K. Nikodem, T. Riedel, and P. K. Sahoo. The stability problem of the Hermite-Hadamard inequality. Math. Inequal. Appl., 1: , 7. Institute of Mathematics, University of Miskolc, H-3515 Miskolc-Egyetemváros, Hungary address: {matjudit,matha}@uni-miskolc.hu