Functional Inequalities Involving Numerical Differentiation Formulas of Order Two

Bulletin of the Malaysian Mathematical Sciences Society, Jan 2017

We write expressions connected with numerical differentiation formulas of order 2 in the form of Stieltjes integral, then we use Ohlin lemma and Levin–Stechkin theorem to study inequalities connected with these expressions. In particular, we present a new proof of the inequality $$\begin{aligned} f\left( \frac{x+y}{2}\right) \le \frac{1}{(y-x)^2} \int _x^y\int _x^yf\left( \frac{s+t}{2}\right) \hbox {d}s\,\hbox {d}t \le \frac{1}{y-x}\int _x^yf(t)\hbox {d}t \end{aligned}$$ satisfied by every convex function \(f{:}\,\mathbb R\rightarrow \mathbb R\) and we obtain extensions of this inequality. Then we deal with non-symmetric inequalities of a similar form.

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Functional Inequalities Involving Numerical Differentiation Formulas of Order Two

Functional Inequalities Involving Numerical Differentiation Formulas of Order Two Tomasz Szostok 0 0 Institute of Mathematics, University of Silesia , Bankowa 14, 40-007 Katowice , Poland We write expressions connected with numerical differentiation formulas of order 2 in the form of Stieltjes integral, then we use Ohlin lemma and LevinStechkin theorem to study inequalities connected with these expressions. In particular, we present a new proof of the inequality f f Communicated by Mohammad Sal Moslehian. Hermite-Hadamard inequality; Differentiation formulas; Convex functions - f (t )dt 1 Introduction Writing the celebrated Hermite–Hadamard inequality f (t )dt ≤ B Tomasz Szostok in the form F (y) − F (x ) y − x y − x Remark 1 Let f, F, : [x , y] → R be such that = F, F = f, let ni , mi ∈ N ∪ {0}, i = 1, 2, 3; ai, j ∈ R, αi, j , βi, j ∈ [0, 1], αi, j + βi. j = 1, i = 1, 2, 3; j = 1, . . . , ni ; bi, j ∈ R, γi, j , δi, j ∈ [0, 1], γi, j + δi, j = 1, i = 1, 2, 3; j = 1, . . . , mi . If the inequality i=1 in=21 a2,i F (α2,i x + β2,i y) y − x in=31 a3,i (α3,i x + β3,i y) im=21 b2,i F (γ2,i x + δ2,i y) (y − x )2 y − x i=1 im=31 b3,i (γ3,i x + δ3,i y) (y − x )2 The simplest expression used to approximate the second-order derivative of f is of the form Remark 2 From numerical analysis it is known that This means that for convex g and for G such that G G(x ) − 2G = g we have + G(y) Lemma 1 (Ohlin [7]) Let X1, X2 be two random variables such that EX1 = EX2 and let F1, F2 be their distribution functions. If F1, F2 satisfy for some x0 the following inequalities E f (X1) ≤ E f (X2) for all continuous and convex functions f : R → R. f (x )d F1(x ) ≤ f (x )d F2(x ) for all continuous and convex functions f : [a, b] → R it is necessary and sufficient that F1 and F2 verify the following three conditions: F1(t )dt ≤ F2(t )dt, x ∈ (a, b), F1(t )dt = F2(t )dt. Let now f : [x , y] → R be any function and let F, : [x , y] → R be such that F = f and = F. We need to write the expression (x ) − 2 in the form where F1 : [0, 1] → R is given by (0) − 2 (1) = f d F1, F1(t ) := −2x 2 + 4x − 1 Proof Let F : [0, 1] → R be such that enough to do the following calculations f d F1 = 4x f (x )dx + (−4x + 4) f (x )dx = 2F − 0 · F (0) 4F (x )dx 4F (x )dx − 0 · F (1) − 2F = 4 (1) − 8 Remark 4 Observe that if equality is satisfied and f are such as in Proposition 1 then the following (x ) − 2 ds dt. After this observation it turns out that inequalities involving the expression (9) were considered in the paper of Dragomir [3] where (among others) the following inequalities were obtained f (t )dt. 2 The Symmetric Case We start with the following remark. f (t )dt F∗(t )dt = Ta f (x , y) = if a ≤ 0, then if a ≤ 2, then if a ≥ 6, then if a ≥ −6, then f (t )dt, f (t )dt, Ta f (x , y) ≤ f Ta f (x , y) ≤ ≤ Ta f (x , y), Furthermore, if a ∈ (2, 6), then the expressions Ta f (x , y), f x+y are not comparable in the 2 class of convex functions, if a < −6, then the expressions Ta f (x , y), f (x)+ f (y) are not comparable in the 2 class of convex functions. F1(t ) := f d F1 = f d F1 = + 1 2 a = 1 − 2 (F (1) − F (0)) + 2a dt = F − F (0) − 2 F (t )2adt ϕ(s) := F3(t ) − F1(t )dt, s ∈ [0, 1] f d F1 ≤ f d F2, F3(t ) := F4(t )dt This theorem provides us with a full description of inequalities which may be obtained using Stieltjes integral with respect to a function of the form (17). Some of the obtained inequalities are already known. For example from (12) and (13) we obtain the inequality (y − x )2 f (t )dt, ds dt. (y − x )2 x f (t )dt ≤ 3 y − x x (y − x )2 x y − x x (y − x )2 x (y − x )2 x (y − x )2 x y − x x In Corollary 1 we obtained inequalities for the triples: (y − x )2 (y − x )2 f (t )dt, f (t )dt, f (y − x )2 Remark 7 Using the functions: F1 defined by (10) and F5 given by F5(t ) := t = 0, t ∈ 0, 21 t ∈ 21 , 1 t = 1, we can see that 1 1 + 6 f (y) ≥ (y − x )2 3 The Non-symmetric Case Now, in contrast to the symmetric case (Remark 5), it is possible to prove inequalities using just one quadratic function but before we do this we shall present a non-symmetric version of Hermite–Hadamard inequality involving only the primitive function of f. Proposition 2 Let x , y be some real numbers such that x < y and let α ∈ [0, 1]. Let f : [x , y] → R, be a convex function, let F : [x , y] → R be such that F = f. If Sα1 f (x , y) is defined by y − x then the following inequality is satisfied: Proof As usually, the proof will be done on the interval [0, 1]. Define the functions F6, F7, F8 : [0, 1] → R by the following formulas: F6(t ) := F8(t ) := F6(t )dt = F7(t )dt = f d F6 = f (1 − α), f d F7 = α f (0) + (1 − α) f (1) f d F8 = Sα1 f (0, 1). (4 − 6α)F (y) + (2 − 6α)F (x ) y − x (6 − 12α)( (y) − (y − x )2 then the following conditions hold true: if α ∈ 31 , 23 then Sα2 f (x , y) ≤ α f (x ) + (1 − α) f (y), Sα2 f (x , y) ≥ f (αx + (1 − α)y), Sα2 f (x , y) ≤ Sα1 f (x , y) and if α ∈ 31 , 21 ∪ 21 , 23 then Sα1 f (x , y) and Sα2 f (x , y) are incomparable in the class of convex functions. Proof Take and let F6, F7, F8 be defined so as in Proposition 2. Then we have f d F1 = (6 − 12α)t + 6α − 2 f (t )dt (1) − F9(t )dt = F8(t )dt = F7(t )dt = F6(t )dt. f d F9 ≤ f d F7, Thus for every convex function f we have f d F6 ≤ f d F9, F9(t )dt = F6(t )dt f d F6, 4 Concluding Remarks and Examples F1(t ) := ax 2 + (1 − α)x cx 2 + (1 − cα − c)x + cα where c = − 1−αα , must be used. Since the description of all possible cases in Theorem 2 was already quite complicated, we shall not present these inequalities in details here. F1(t ) := t ≤ 41 , f (t )dt, 8 (x ) − 16 − 16 (y − x )2 where f : [x , y] → R is any convex function and = f. Remark 10 We have t 2d F6(t ) = (1 − α)2. This means that for two values of α : 3−6√3 and 3+6√3 we have 01 t 2d F9(t ) = 01 t 2d F6(t ). Moreover, as it was mentioned in the proof of Theorem 3, the functions F9, F6 have in this case two crossing points. This implies (see [2,9]) that the inequalities: 3 − √3 √3 3 − are satisfied by all 2-convex functions f : [x , y] → R Remark 11 It is easy to see that all inequalities obtained in this paper in fact characterize convex functions (or 2-convex functions). This is a consequence of results contained in paper [1]. 1. Bessenyei , M. , Páles, Zs: Characterization of higher order monotonicity via integral inequalities . Proc. R. Soc. Edinb. Sect. A 140 ( 4 ), 723 - 736 ( 2010 ) 2. Denuit , M. , Lefevre , C. , Shaked , M. : The s-convex orders among real random variables, with applications . Math. Inequal. Appl. 1 ( 4 ), 585 - 613 ( 1998 ) 3. Dragomir , S.S. : Two mappings in connection to Hadamard's inequalities . J. Math. Anal. Appl. 1(167) , 49 - 56 ( 1992 ) 4. Dragomir , S.S. , Gomm, I.: Some new bounds for two mappings related to the Hermite-Hadamard inequality for convex functions . Numer. Algebra Control Optim . 2 ( 2 ), 271 - 278 ( 2012 ) 5. Levin , V.I. , Stecˇkin , S.B. : Inequalities. Am. Math. Soc. Transl . 14 ( 2 ), 1 - 29 ( 1960 ) 6. Niculescu , C.P. , Persson , L.-E. : Convex Functions and Their Applications . A Contemporary Approach. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC , vol. 23 . Springer, New York ( 2006 ) 7. Ohlin , J. : On a class of measures of dispersion with application to optimal reinsurance . ASTIN Bull . 5 , 249 - 266 ( 1969 ) 8. Olbrys´ , A. , Szostok , T.: Inequalities of the Hermite-Hadamard type involving numerical differentiation formulas . Results Math . 67 , 403 - 416 ( 2015 ) 9. Rajba , T.: On the Ohlin lemma for Hermite-Hadamard-Fejer type inequalities . Math. Inequal. Appl . 17 ( 2 ), 557 - 571 ( 2014 ) 10. Szostok , T.: Ohlin's lemma and some inequalities of the Hermite-Hadamard type . Aequ. Math. 89 , 915 - 926 ( 2015 ) 11. Szostok , T.: Levin Steckin Theorem and Inequalities of the Hermite-Hadamard Type. (submitted) arXiv:1411 .7708 [math.CA]


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Tomasz Szostok. Functional Inequalities Involving Numerical Differentiation Formulas of Order Two, Bulletin of the Malaysian Mathematical Sciences Society, 2017, 1-14, DOI: 10.1007/s40840-017-0462-3