docGENERALIZATIONS OF OPIAL-TYPE INEQUALITIES IN SEVERAL
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GENERALIZATIONS OF OPIAL-TYPE INEQUALITIES IN SEVERAL
DEMONSTRATIO MATHEMATICA Vol. 49 No 2 2016 Maja Andrić, Ana Barbir, Josip Pečarić, Gholam Roqia CORRIGENDUM TO “GENERALIZATIONS OF OPIAL-TYPE INEQUALITIES IN SEVERAL INDEPENDENT VARIABLES” PUBLISHED IN DEMONSTRATIO MATH. 4(47) (2014), 324–335 Communicated by A. Fryszkowski Abstract. The purpose of this corrigendum is to correct an error in the earlier paper by the authors: Generalizations of Opial-type inequalities in several independent variables, Demonstratio Math. In the paper “Generalizations of Opial-type inequalities in several independent variables” published in Demonstratio Mathematica ([1]), we have considered certain multidimensional Opial-type inequalities, and for two of them, inequalities obtained in Theorem 2.1 (page 841) and Theorem 2.3 (page 844), we give corrigendum. Namely, the error was made in a final step of the proof of Theorem 2.1, in the equality (2.7). Here we made necessary ś corrections, which resulted from the need to observe the inequality on Ω“ m j“1 raj , bj s with boundary conditions only in a “ pa1 , . . . , am q. Since applied Theorem 2.1 was used in Theorem 2.3, we made appropriate changes in Theorem 2.3, also. Followingś notation is used: ś Let Ω “ m volpΩq “ m j“1 raj , bj s and ś j“1 pbj ´ aj q. Let t “ pt1 , . . . , tm q be a general point in Ω, Ωt “ m ra , t s j“1 j j and dt “ dt1 . . . dtm . Further, let Dupxq “ d upxq, dx Dk upt1 , . . . , tm q “ B upt1 , . . . , tm q Btk and Dk upt1 , . . . , tm q “ D1 ¨ ¨ ¨ Dk upt1 , . . . , tm q, 2010 Mathematics Subject Classification: Primary 26D10; Secondary 26B25. Key words and phrases: Opial-type inequalities, Willett’s inequality, Rozanova’s inequality, several independent variables. DOI: 10.1515/dema-2016-0013 c Copyright by Faculty of Mathematics and Information Science, Warsaw University of Technology 150 M. Andrić, A. Barbir, J. Pečarić, G. Roqia 1 ≤ k ≤ m. Let Ω1 “ śm j“2 raj , bj s and dt1 “ dt2 . . . , dtm . Let Djl upt1 , . . . , tm q “ B jl upt1 , . . . , tm q, Btlj . . . Btl1 1 ≤ j ≤ m, 1 ≤ l ≤ n. Also, by C mn pΩq we denote the space of all functions u on Ω which have continuous derivatives Djl u for j “ 1, . . . , m and l “ 1, . . . , n. Proofs of corrected theorems follow the same step as in [1], but finish with the inequality using boundary conditions only in a. First, we give corrigendum to [1, Theorem 2.1]. Notice that the equation (2.2) from [1] is explained in more detail here. Theorem 1. Let m, n, p P N. Let f be a nonnegative and differentiable function on r0, 8qp , with f p0, . . . , 0q “ 0. Further, for i “ 1, . . . , p let xi P C mn pΩq be such that Djl xi ptq|tj “aj “ 0, where j “ 1, . . . , m and l “ 0, . . . , n ´ 1. Also, let Di f , i “ 1, . . . , p, be nonnegative, continuous and nondecreasing on r0, 8qp . Then the following inequality holds ˙ ż ˆÿ p (1) Di f p|x1 ptq| , . . . , |xp ptq|q |Dmn xi ptq| dt Ω i“1 ˆ ż pn ´ 1q!m pvolpΩqqn´1 |Dmn x1 ptq| dt, . . . , ≤ f pn ´ 1q!m pvolpΩqqn´1 Ω ˙ ż pvolpΩqqn´1 mn |D xp ptq| dt . pn ´ 1q!m Ω Proof. We extend technique used in [2, Theorem 2.1] on a multidimensional case. For continuous function g : Ω Ñ R, we should define y : Ω Ñ R such that B mn y (2) Dmn ypx1 , . . . , xm q “ n “ gpx1 , . . . , xm q Bxm ¨ ¨ ¨ Bxn1 and ż ź m 1 (3) ypx1 , . . . , xm q “ pxj ´ tj qn´1 gpt1 , . . . , tm q dt1 ¨ ¨ ¨ dtm , pn ´ 1q!m j“1 Ωx where Ωx “ Define śm (4) ypxq “ j“1 raj , xj s. ż t1 żx dt1 a ż tn´2 dt2 ¨ ¨ ¨ a ż tn´1 dtn´1 a gptn q dtn a Corrigendum to “Generalizations of Opial-type inequalities” or, in different notations ż ż ż 2 1 dt ¨ ¨ ¨ (5) ypxq “ dt Ωt1 Ωx 151 ż gptn q dtn , n´1 dt Ωtn´2 Ωtn´1 where a “ pa1 , . . . , am q, x “ px1 , . . . , xm q, ti “ pti1 , . . . , tim q, dti “ dti1 ¨ ¨ ¨ dtim , i “ 1, . . . , n and Ωti “ m ź raj , tij s, Ωti Ď Ωti´1 , i “ 1, . . . , n ´ 1. j“1 Since g is a continuous function, (2) obviously follows. Obviously, integrals on the right-hand side of (4) or (5), can be written as iterations of the integrals of the form ż xj ż t1 ż tn´2 ż tn´1 j j j n´1 1 2 dtj dtj ¨ ¨ ¨ dtj grptnj q dtnj , aj aj aj aj which are known (and easy to deduce by interchanging the order of integration) to be equal to ż xj 1 pxj ´ tnj qn´1 grptnj q dtnj , pn ´ 1q! aj j “ 1, . . . , m, from which (3) easily follows. Let ż ź m 1 (6) yi ptq “ ptj ´ sj qn´1 |Dmn xi psq| ds, pn ´ 1q!m j“1 Ωt for t P Ω, i “ 1, . . . , p. Hence Dmn yi ptq “ |Dmn xi ptq| and yi ptq ≥ |xi ptq|. It is easy to conclude that for each l “ 0, . . . , n ´ 1 we have Djl yi ptq ≥ 0 and nondecreasing on Ω (i “ 1, . . . , p and j “ 1, . . . , m). From Djl yi ptq|tj “aj “ 0 follows pvolpΩqqn´1 mpn´1q yi ptq ≤ D yi ptq, t P Ω. pn ´ 1q!m Define pvolpΩqqn´1 mpn´1q ui ptq “ D yi ptq pn ´ 1q!m for t P Ω and i “ 1, . . . , p. Since Di f are nonnegative, continuous and 152 M. Andrić, A. Barbir, J. Pečarić, G. Roqia nondecreasing on r0, 8qp , it follows ż „ÿ p mn (7) Di f p|x1 ptq| , . . . , |xp ptq|q |D xi ptq| dt i“1 Ω ≤ ż „ÿ p Ω and ż «ÿ p Ω Di f py1 ptq, . . . , yp ptqq D mn yi ptq dt, i“1 ff Di f py1 ptq, . . . , yp ptqq Dmn yi ptq dt i“1 ż „ÿ p pvolpΩqqn´1 mpn´1q D y1 ptq, . . . , pn ´ 1q!m i“1 ˙ Ω pvolpΩqqn´1 mpn´1q mn D yp ptq D yi ptq dt pn ´ 1q!m ż b1 „ ÿ ż p mn 1 ≤ Di f pu1 pt1 , b2 , . . . , bm q, . . . , up pt1 , b2 , . . . , bm qq ˆ D yi ptq dt dt1 ≤ a1 ≤ ˆ Di f i“1 ż b1 „ ÿ p a1 Ω1 Di f pu1 pt1 , b2 , . . . , bm q, . . . , up pt1 , b2 , . . . , bm qq i“1 pn ´ 1q!m D1 ui pt1 , b2 . . . , bm q dt1 ˆ pvolpΩqqn´1 ż b1 pn ´ 1q!m d rf pu1 pt1 , b2 , . . . , bm q, . . . , up pt1 , b2 , . . . , bm qqs dt1 “ pvolpΩqqn´1 a1 dt1 pn ´ 1q!m f pu1 pb1 , b2 , . . . , bm q, . . . , up pb1 , b2 , . . . , bm qq “ pvolpΩqqn´1 pn ´ 1q!m f “ pvolpΩqqn´1 ˆ ˙ ż ż pvolpΩqqn´1 pvolpΩqqn´1 mn mn ˆ |D x1 ptq| dt, . . . , |D xp ptq| dt . pn ´ 1q!m pn ´ 1q!m Ω Ω Next comes a result for a convex function f . The proof follows the same steps as in [1, Theorem 2.3], again with the difference of observing the inequality on Ω with boundary conditions only in a. We will use the following lemma about convex function of several variables ([3, page 11]). Lemma 1. Suppose that f is defined on the open convex set U Ă Rn . If f is pstrictlyq convex on U and the gradient vector f 1 pxq exists throughout U , then f 1 is pstrictlyq increasing on U . Corrigendum to “Generalizations of Opial-type inequalities” 153 Theorem 2. Let m, n, p P N. Let f be a convex and differentiable function on r0, 8qp with f p0, . . . , 0q “ 0. Further, for i “ 1, . . . , p let xi P C mn pΩq be such that Djl xi ptq|tj “aj “ 0, where j “ 1, . . . , m and l “ 0, . . . , n ´ 1. Then the following inequality holds ż ´ÿ p ¯ (8) Di f p|x1 ptq| , . . . , |xp ptq|q |Dmn xi ptq| dt Ω i“1 ≤ pn ´ 1q!m pvolpΩqqn ˆ ż f pvolpΩqqn mn |D x1 ptq| , . . . , pn ´ 1q!m Ω ˙ pvolpΩqqn mn |D xp ptq| dt. pn ´ 1q!m Proof. As in the proof of the previous theorem, we obtain p1q with the difference of applying Lemma 1 in p7q since f is a convex function. Then, from Jensen’s inequality for integrals (see for ff example [3, page 51]), we have ˆ ˙ ż «ÿ p Di f |x1 ptq| , . . . , |xp ptq| |Dmn xi ptq| dt Ω i“1 ˆ ż pn ´ 1q!m pvolpΩqqn´1 ≤ |Dmn x1 ptq| dt, . . . , n´1 f m pn ´ 1q! pvolpΩqq Ω pvolpΩqqn´1 pn ´ 1q!m ż |D mn ˙ xp ptq| dt Ω ˆ ż 1 pvolpΩqqn mn pn ´ 1q!m |D x1 ptq| dt, . . . , f “ pvolpΩqq pn ´ 1q!m pvolpΩqqn´1 Ω ˙ ż 1 pvolpΩqqn mn |D xp ptq| dt pvolpΩqq pn ´ 1q!m Ω ˙ ż ˆ n m pn ´ 1q! pvolpΩqq pvolpΩqqn mn mn ≤ f |D x1 ptq| , . . . , |D xp ptq| dt. pvolpΩqqn pn ´ 1q!m pn ´ 1q!m Ω References [1] M. Andrić, A. Barbir, J. Pečarić, G. Roqia, Generalizations of Opial-type inequalities in several independent variables, Demonstratio Math. 4(47) (2014), 324–335. [2] M. Andrić, A. Barbir, J. Pečarić, On Willett’s, Godunova-Levin’s and Rozanova’s Opial-type inequalities with related Stolarsky type means, Math. Notes 96(6) (2014), 841–854. 154 M. Andrić, A. Barbir, J. Pečarić, G. Roqia [3] J. E. Pečarić, F. Proschan, Y. C. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Inc., 1992. M. Andrić, A. Barbir FACULTY OF CIVIL ENGINEERING, ARCHITECTURE AND GEODESY UNIVERSITY OF SPLIT MATICE HRVATSKE 15 21000 SPLIT, CROATIA E-mail: [email protected], [email protected] J. Pečarić FACULTY OF TEXTILE TECHNOLOGY UNIVERSITY OF ZAGREB PRILAZ BARUNA FILIPOVIĆA 28A 10000 ZAGREB, CROATIA E-mail: [email protected] G. Roqia ABDUS SALAM SCHOOL OF MATHEMATICAL SCIENCES 68-B NEW MUSLIM TOWN LAHORE 54000, PAKISTAN E-mail: [email protected] Received July 20, 2015.
