Miskolc Mathematical Notes HU e-issn On bivariate Meyer-König and Zeller operators. Ali Olgun

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1 Misolc Mathematical Notes HU e-issn Vol. 4 (03), No 3, pp OI: 0.854/MMN On bivariate Meyer-König and Zeller operators Ali Olgun

2 Misolc Mathematical Notes HU e-issn Vol. 4 (03), No. 3, pp ON BIVARIATE MEYER-KÖNIG AN ZELLER OPERATORS ALI OLGUN Received 3 May, 0 Abstract. This wor relates to bivariate Meyer-König and Zeller operators, M n n N which are not a tensor product setting. We show the monotonicity o the sequence o operators or n under convexity, moreover we study the property o monotonicity in the sense o Li [9]. Finally, we provide an rth order generalization M Œr n 00 Mathematics Subject Classiication: 4A5 4A36 o M n and also study approximation o M Œr n. Keywords: multivariate Meyer-König and Zeller operator, convexity, monotonicity, modulus o continuity. INTROUCTION The Cheney and Sharma modiication o the well-nown univariate Meyer-König and Zeller operators (MKZ) are deined as Mn.x/ X m n.x/ 0 n C n N (.) or C Œ0/ and x Œ0/ n N where m n.x/. x/ nc n C x [3]. The monotonic convergence o Mn. Ix/g n under convexity was investigated in [] by Cheney and Sharma by means o analytical approach, also studied in [8] by Khan, by means o probabilistic approach. Monotonic convergence o several approximation operators are studied deeply in [6] by Khan, ella-vecchia and Fassih with probabilistic point o view. Some wors related to MKZ operators can be viewed in [ 8, 3] and [0, ]. We shall use the ollowing standard notation. Let x.x x /, t.t t / R. / N 0, N 0 N[0g: x x :x Š Š: Š jj C jxj c 03 Misolc University Press

3 0 ALI OLGUN x C x e i denotes the unit vector in R : Furthermore P 0 P P. 0 0 Let S R be the open simplex deined by n nš Š.n jj/š, and S x R I x i 0 i jxj < : For a unction C.S /, bivariate MKZ operators are deined as M n.x/. X jxj/ nc n C jj x n N (.) n C jj 0 where C.S / denotes the space o continuous real valued unctions deined on S : It is obvious that M n n N are not a tensor product extension o the univariate MKZ operators Mn given by (.). Now we give the ollowing deinitions which we shall use. einition. A continuous unction is said to be convex in R m i rx ix i i rx i.x i / or every x x :::x r and or every non-negative numbers ::: r such that C C ::: C r. einition. A continuous unction rom R into R is said to be Lipschitz continuous o order.0 i there exists a constant A > 0 such that or every.x x /.y y / satisies j.x x / i.y y /j A X jx i y i j i the set o Lipschitz continuous unctions is denoted by Lip A./: In this wor, we irstly show the monotonicity o the sequence o bivariate MKZ operators deined by (.) under convexity. Secondly we give a ind o monotonnicity similar to the property given by Li in [9]. Namely, we show that i.x/ is a nonnegative unction and x.x/.i / is non-increasing or x i on.0/ then or each n xi build an r i M n. Ix/ is also non-increasing or x i on.0/. Moreover we th order generalization Mn Œr construction in [8] and investigate its approximation property. o M n analogues to Kirov and Popova s

4 ON BIVARIATE MEYER-KÖNIG AN ZELLER OPERATORS 03. MONOTONICITY FOR THE SEQUENCE OF BIVARIATE MEYER-KÖNIG AN ZELLER OPERATORS In this section, we study the monotonic convergence o the sequence o bivariate MKZ operator under convexity. Note that monotonic convergence o univariate MKZ operator when is convex, was irst obtained by Cheney and Sharma in [3]. We note here that monotonic convergence o the multivariate Basaov operator is studied in []. Theorem. I is convex, then M n. Ix/ is strictly monotonically non-decreasing in n unless is the linear unction (in which case M n. Ix/ M nc. Ix/ or all n N/. Proo. We have M n. Ix/ M nc. Ix/. jxj/ nc 8 " < X # n C n C C 0 0 : n C n C C " X # n C n C C C 0 0 n C n C C C C X X 0 0 X X 0 0 n C C jj n C C jj n C C jj n C C jj Thereore the last equation reduces to the ollowing: x C x x x C 9 = : M n. Ix/ M nc. Ix/. jxj/ nc 8 " < X n C n C C 0 0 : n C n C C C 0 # n C " x n C C X n C 0 n C # n C C n C 0 C 0 n C C n C " X X C n C jj n C jj n C jj e x x x

5 04 ALI OLGUN C n C jj n C jj n C jj e ) n C jj n C C jj C #x : n C jj n C C jj Let I, I and I 3 denote the ollowings n C jj n C C jj I n C jj n C C jj e n C jj e n C jj C C n C jj e n C jj e n C I 0 n C C 0 n C n C n C C 0 n C C n C n C C I n C n C C n C C 0 : n C For I, we tae n C jj n C C jj, n C jj e n C C jj, 3 n C jj e n C C jj. Clearly and 3 are non-negative numbers, and C C 3 : On the other hand i we set x n C jj x e n C jj x 3 e n C jj then it ollows that x C x C 3x 3 n C C jj :

6 ON BIVARIATE MEYER-KÖNIG AN ZELLER OPERATORS 05 Hence, rom the convexity o we obtain that I 0: For I we set n C n C n C C n C C and y 0 y 0 n C n C then, clearly, 0 and C : Morever we have y C y 0 : n C Thus, rom the convexity o we obtain that I 0: Similarly we deduce that I 3 0: So we have proved M n. Ix/ M nc. Ix/ or all n N: Using the similar arguments given in [], the last part o the proo is given as ollows: The equality o M n. Ix/ and M nc. Ix/ can be satisied only i I 0 or all N and I I 3 0 or all N 0 : But, i is convex, then I 0 implies that its graph is a plane in triangle. In the ollowing, we show that the bivariate MKZ operators can retain a certain monotony, that is Theorem. Let be deined on S : I.x/ is a non-negative unction such that.x/ x i.i / is non-increasing or x i on.0/ then or each n M n. Ix/ x i is also non-increasing or x i on.0/: Proo. Straightorward computation gives that or n we have, or example with respect to Mn. x < X X. x x / nc 9 n C C = : x n C C n C C C x 0 0 < X. x x / nc n C 0 : x n C C X X 0 0 ). x x / nc n C C x x n C C n C C C X x x / nc n C x n C x C X X 0

7 06 ALI OLGUN n C h i. x x / nc x x n C C n C C X.n C /. x x / n x. x x / nc n C x 0 0 n C X X n C C C x n C C n C C C 0 h i.n C /. x x / n x C. x x / nc. /x X. x x / nc.n C /x n C x 0 x 0 x x n C X X n C C C. x x / n x n C C n C C C 0 h i nx x C. /x. /x. /x x x. x /. x x / n x X. x x / nc. C nx x / 0 x x C x n C X n C C n C C n C C C i.n C /x C. /. x /x 0 X 0 n C x. x x / n x h X. x x / n n C x.. C nx x // 0 x 0 n C X X n C C C x. n /x n C C n C C C 0 X X. x x / n n C C C n C C n C C C 0 X. C nx x / n C x 0. x x / n x 0 n C X X n C C x.n C /x n C C n C C C 0 x

8 . x x / n C ON BIVARIATE MEYER-KÖNIG AN ZELLER OPERATORS 07 X 0 X C n C C C n C C n C C C C. x /x. x x / n x X. C nx x / n C x 0. x x / n x 0 n C X X C C n C C C n C C n C C C n C C C C C C x 0 X X. x /. x x / n x x n C C n C C n C C C.n C /. x x / n x x x 0 Since or 0 and n C C C C C 0 < n C C C C n C C < n C C nx x 0 x or x x S n N so we have or Mn. x X. C nx x / n C 0. x x / n x n C C X X 0 x n C C C C n C C C n C C C.n C C / n C C n C C. x x / n x x x n C C C C C is non-positive, since is a non-negative unction, such that.x/ x i.i / is nonincreasing or x i on.0/: Similar calculations can be obtained or x :

9 08 ALI OLGUN 3. A GENERALIZATION This section provides an r th order generalization o the bivariate MKZ operators in the sense o Kirov and Popova s construction [8]. Let C r.s / r N 0, denote the space o all unctions deined on S and having all continuous partial derivatives up to order r. By Mn Œr we denote the ollowing generalization o M n. For xt S Mn Œr. Ix/ M n.x / (3.) where r P r ncjj x W.x x / x n C jj x n C jj x n @x.x:r/ r W X r.x / i.x / r (3.) icj r j are binomial coeicients, and P r.x / ncjj n C jj C.x:r/ Š C.x:r/ n C jj n C jj C ::: C.x:r/r rš the Taylor polynomial or at n C jj S : Now we state the ollowing pointwise estimate or Mn Œr Theorem 3. I C r.s ˇ ˇMn Œr. Ix/.x/ˇˇˇ A.r /Š n C jj (3.3) Lip A./ i C j r then we have C r B.r/M n.gix/ (3.4) or x S where g.s/ jx sj rc B.r/ is the amiliar beta unction, r n N 0 0 < and A > 0: Proo. From (3.) and (3.3) we have.x/ M Œr n. Ix/ X R r ncjj 0.x / n C jj x. jxj/ nc (3.5)

10 where R r ncjj ON BIVARIATE MEYER-KÖNIG AN ZELLER OPERATORS 09.x / W.x/ (3.6) can be given by R r ncjj rx.x:r/ h n C jj x. jxj/ nc : hš n C jj h0.x /.r /Š Using (3.),(3.7) results in R r ncjj.x /.r /Š Z 0.x:r/ r n C jj C tx Z 0 X icj r n C jj C tx n C jj Substituting (3.8) into (3.5) and taing into account that at the ollowing. ˇ ˇ.x/ Mn Œr. Ix/ˇˇˇ A.r /Š which proves (3.4). r.x / i.x / r n C jj X.jx j C jx j/ r 0 (3.6). t/ r dt: (3.7). t/ r dt: Lip A./ we arrive jx j C jx j n C jj Z x. jxj/ nc t. t/ r dt 0 A X.jx j C jx j/ rc n C jj x. jxj/ nc B. C r/.r /Š 0 A.r /Š C r B.r/ X ˇ rc n C jj ˇx n C jjˇ x. jxj/ nc For the uniorm convergence o Mn Œr. / let us consider the above mentioned unction g.s/ jx sj rc. Obviously g.x/ 0 and g is continuous on S : From multivariate extension o the Bohman-Korovin theorem (see []) we have 0 M n.g/ C.S / 0 as n :

11 030 ALI OLGUN Thereore by means o Theorem 3 we arrive at the ollowing result Mn Œr. / C 0 as n :.S / REFERENCES [] A. Altın, O. oğru, and F. Taşdelen, The generalization o Meyer-König and Zeller operators by generating unctions, J. Math. Anal. Appl., vol. 3, no., pp. 8 94, 005. [] F. Cao, C. ing, and Z. Xu, On multivariate Basaov operator, J. Math. Anal. Appl., vol. 307, no., pp. 74 9, 005. [3] E. W. Cheney and A. Sharma, Bernstein power series, Can. J. Math., vol. 6, pp. 4 5, 964. [4] O. oğru and O. uman, Statistical approximation o Meyer-König and Zeller operators based on q-integers, Publ. Math., vol. 68, no. -, pp. 99 4, 006. [5] O. oğru and V. Gupta, Korovin-type approximation properties o bivariate q-meyer-könig and Zeller operators, Calcolo, vol. 43, no., pp. 5 63, 006. [6] M. Khan, B. ella Vecchia, and A. Fassih, On the monotonicity o positive linear operators, J. Approx. Theory, vol. 9, no., pp. 37, art. no. at963 3, 998. [7] R. A. Khan, Some probabilistic methods in the theory o approximation operators, Acta Math. Acad. Sci. Hung., vol. 35, pp , 980. [8] G. Kirov and L. Popova, A generalization o the linear positive operators, Math. Bal., New Ser., vol. 7, no., pp. 49 6, 993. [9] Z. Li, Bernstein polynomials and modulus o continuity, J. Approx. Theory, vol. 0, no., pp. 7 74, art. no. jath , 000. [0] A.-J. López-Moreno and F.-J. Muñoz-elgado, Asymptotic expansion o multivariate conservative linear operators, J. Comput. Appl. Math., vol. 50, no., pp. 9 5, 003. [] T. Tri, An elementary proo o the preservation o Lipschitz constants by the Meyer-König and Zeller operators, JIPAM, J. Inequal. Pure Appl. Math., vol. 4, no. 5, p. 3, 003. [] V. I. Volov, On the convergence o sequences o linear positive operators in the space o continuous unctions o two variables, ol. Aad. Nau SSSR, vol. 5, pp. 7 9, 957. [3] H. Wang, Properties o convergence or the q-meyer-könig and Zeller operators, J. Math. Anal. Appl., vol. 335, no., pp , 007. Author s address Ali Olgun Kırıale University, Faculty o Art and Science, epartment o Mathematics, 7450, Yahşihan, Kırıale, Turey address: aliolgun7@gmail.com