n-points Inequalities of Hermite-Hadamard Type for h-convex Functions on Linear Spaces

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1 Armenin Journl o Mthemtics Volume 8, Number, 6, n-points Inequlities o Hermite-Hdmrd Tpe or h-convex Functions on Liner Spces S. S. Drgomir Victori Universit, Universit o the Witwtersrnd Abstrct. Some n-points inequlities o Hermite-Hdmrd tpe or h-convex unctions deined on convex subsets in rel or complex liner spces re given. Applictions or norm inequlities re provided s well. Ke Words: Convex unctions, Integrl inequlities, h-convex unctions. Mthemtics Subject Clssiiction : 6D5; 5D Introduction We recll here some concepts o convexit tht re well known in the literture. Let I be n intervl in R. Deinition ([6]) We s tht : I R is Godunov-Levin unction or tht belongs to the clss Q (I) i is non-negtive nd or ll x, I nd t (, ) we hve (tx + ( t) ) t (x) + (). () t Some urther properties o this clss o unctions cn be ound in [], [], [3], [3], [35] nd [36]. Among others, its hs been noted tht nonnegtive monotone nd non-negtive convex unctions belong to this clss o unctions. The bove concept cn be extended or unctions : C X [, ) where C is convex subset o the rel or complex liner spce X nd the inequlit () is stisied or n vectors x, C nd t (, ). I the unction : C X R is non-negtive nd convex, then is o Godunov- Levin tpe. 38

2 n-points Inequlities o Hermite-Hdmrd Tpe 39 Deinition ([3]) We s tht unction : I R belongs to the clss P (I) i it is nonnegtive nd or ll x, I nd t [, ] we hve (tx + ( t) ) (x) + (). () Obviousl Q (I) contins P (I) nd or pplictions it is importnt to note tht lso P (I) contins ll nonnegtive monotone, convex nd qusi convex unctions, i. e. nonnegtive unctions stising (tx + ( t) ) mx { (x), ()} (3) or ll x, I nd t [, ]. For some results on P -unctions see [3] nd [33] while or qusi convex unctions, the reder cn consult []. I : C X [, ), where C is convex subset o the rel or complex liner spce X, then we s tht it is o P -tpe (or qusi-convex) i the inequlit () (or (3)) holds true or x, C nd t [, ]. Deinition 3 ([7]) Let s (, ]. A unction : [, ) [, ) is sid to be s-convex (in the second sense) or Breckner s-convex i (tx + ( t) ) t s (x) + ( t) s () or ll x, [, ) nd t [, ]. For some properties o this clss o unctions see [], [], [7], [8], [8], [9], [7], [9] nd [38]. The concept o Breckner s-convexit cn be similrl extended or unctions deined on convex subsets o liner spces. It is well known tht i (X, ) is normed liner spce, then the unction (x) = x p, p is convex on X. Utilising the elementr inequlit ( + b) s s + b s tht holds or n, b nd s (, ], we hve or the unction g (x) = x s tht g (tx + ( t) ) = tx + ( t) s (t x + ( t) ) s (t x ) s + [( t) ] s = t s g (x) + ( t) s g () or n x, X nd t [, ], which shows tht g is Breckner s-convex on X. In order to uni the bove concepts or unctions o rel vrible, S. Vrošnec introduced the concept o h-convex unctions s ollows. Assume tht I nd J re intervls in R, (, ) J nd unctions h nd re rel non-negtive unctions deined in J nd I, respectivel.

3 4 S. S. Drgomir Deinition 4 ([4]) Let h : J [, ) with h not identicl to. We s tht : I [, ) is n h-convex unction i or ll x, I we hve or ll t (, ). (tx + ( t) ) h (t) (x) + h ( t) () (4) For some results concerning this clss o unctions see [4], [6], [3], [39], [37] nd [4]. This concept cn be extended or unctions deined on convex subsets o liner spces in the sme w s bove replcing the intervl I be the corresponding convex subset C o the liner spce X. We cn introduce now nother clss o unctions. Deinition 5 We s tht the unction : C X [, ) is o s-godunov-levin tpe, with s [, ], i or ll t (, ) nd x, C. (tx + ( t) ) t s (x) + ( t) s (), (5) We observe tht or s = we obtin the clss o P -unctions while or s = we obtin the clss o Godunov-Levin. I we denote b Q s (C) the clss o s-godunov-levin unctions deined on C, then we obviousl hve P (C) = Q (C) Q s (C) Q s (C) Q (C) = Q (C) or s s. The ollowing inequlit holds or n convex unction deined on R ( ) + b b () + (b) (b ) < (x)dx < (b ),, b R. (6) It ws irstl discovered b Ch. Hermite in 88 in the journl Mthesis (see [3]). But this result ws nowhere mentioned in the mthemticl literture nd ws not widel known s Hermite s result. E. F. Beckenbch, leding expert on the histor nd the theor o convex unctions, wrote tht this inequlit ws proven b J. Hdmrd in 893 [5]. In 974, D. S. Mitrinović ound Hermite s note in Mthesis [3]. Since (6) ws known s Hdmrd s inequlit, the inequlit is now commonl reerred s the Hermite-Hdmrd inequlit. For relted results, see []-[3], [4] nd [34]. We cn stte the ollowing generliztion o the Hermite-Hdmrd inequlit or h-convex unctions deined on convex subsets o liner spces [7].

4 n-points Inequlities o Hermite-Hdmrd Tpe 4 Theorem Assume tht the unction : C X [, ) is h-convex unction with h L [, ]. Let, x C with x nd ssume tht the mpping [, ] t [( t) x + t] is Lebesgue integrble on [, ]. Then ( ) x + h ( ) [( t) x + t] dt [ (x) + ()] h (t) dt. (7) Remrk I : I [, ) is h-convex unction on n intervl I o rel numbers with h L [, ] nd L [, b] with, b I, < b, then rom (7) we get the Hermite-Hdmrd tpe inequlit obtined b Srik et l. in [37] ( ) + b h ( ) b b (u) du [ () + (b)] h (t) dt. I we write (7) or h (t) = t, then we get the clssicl Hermite-Hdmrd inequlit or convex unctions ( ) x + (x) + () [( t) x + t] dt. (8) I we write (7) or the cse o P -tpe unctions : C [, ), i.e., h (t) =, t [, ], then we get the inequlit ( ) x + [( t) x + t] dt (x) + (), (9) tht hs been obtined or unctions o rel vrible in [3]. I is Breckner s-convex on C, or s (, ), then b tking h (t) = t s in (7) we get ( ) x + s [( t) x + t] dt (x) + (), () s + tht ws obtined or unctions o rel vrible in [8]. Since the unction g (x) = x s is Breckner s-convex on on the normed liner spce X, s (, ), then or n x, X we hve x + s ( t) x + t s dt x s + x s. () s + I : C [, ) is o s-godunov-levin tpe, with s [, ), then ( ) x + (x) + () [( t) x + t] dt. () s+ s

5 4 S. S. Drgomir We notice tht or s = the irst inequlit in () still holds, i.e. ( ) x + 4 [( t) x + t] dt. (3) The cse o unctions o rel vribles ws obtined or the irst time in [3]. Motivted b the bove results, in this pper some n-points inequlities o Hermite-Hdmrd tpe or h-convex unctions deined on convex subsets in rel or complex liner spces re given. Applictions or norm inequlities re provided s well. Some New Results In [7] we lso obtined the ollowing result: Theorem Assume tht the unction : C X [, ) is n h-convex unction with h L [, ]. Let, x C with x nd ssume tht the mpping [, ] t [( t) x + t] is Lebesgue integrble on [, ]. Then or n λ [, ] we hve the inequlities { [ ] [ ]} ( λ) x + (λ + ) ( λ) x + λ h ( ) ( λ) + λ (4) [( t) x + t] dt [ (( λ) x + λ) + ( λ) () + λ (x)] {[h ( λ) + λ] (x) + [h (λ) + λ] ()} We cn stte the ollowing new corollr s well: h (t) dt h (t) dt. Corollr With the ssumptions o Theorem we hve h ( ) { [ ] [ ( λ) x + (λ + ) ( λ) + (λ + ) x ( λ) + [( t) x + t] dt [ ] () + (x) (( λ) x + λ) dλ + h (t) dt [ [ (x) + ()] h (λ) dλ + ] h (t) dt. (5) ]} dλ

6 n-points Inequlities o Hermite-Hdmrd Tpe 43 Proo. The proo ollows b integrting the inequlit (4) over λ nd b using the equlit [ ] ( λ) x + λ λ dλ = The ollowing result or double integrl lso holds: [ ] ( + µ) x + ( µ) ( µ) dµ. Corollr With the ssumptions o Theorem we hve h ( ) (b ) (6) { [ ] [ ]} x + (β + ) (β + ) x + + ddβ () () [( t) x + t] dt [ b b ( βx + (b ) [ b b ( β (b ) h or n b >. Proo. I we tke λ = +β we hve ) ddβ + ) ddβ + ] () + (x) h (t) dt ] [ (x) + ()] h (t) dt, h ( ) { [ ] β βx + () () + [ ]} (β + ) x + () [( t) x + t] dt [ ( ) βx + + β () + ] (x) h (t) dt {[ ( ) β h + ] [ ( ) (x) + h + β ] } () h (t) dt, (7) or n, β with >.

7 44 S. S. Drgomir Since the mpping [, ] t [( t) x + t] is Lebesgue integrble on [, ], then the double integrl b ( ) b βx+ ddβ exists or n b >. +β The sme holds or the other integrls in (6). Integrting the inequlit (7) on the squre [, b] over (, β) we hve h ( ) (b ) { [ ] β βx + () () [ (b ) Observe tht + [ (β + ) x + () ]} ddβ [( t) x + t] dt ( ) βx + + β () + ] (x) ddβ h (t) dt {[ ( ) β h (t) dt h + ] (x) + [ ( ) + h + β ] } () ddβ. (8) = [ β βx + () () [ x + (β + ) () ] ddβ ] ddβ nd then { [ ] β βx + () + [ (β + ) x + () () { [ ] [ x + (β + ) (β + ) x + = + () () Also nd since then we hve ddβ + ddβ = β ddβ = β ddβ ddβ = (b ). ]} ddβ ]} ddβ. ddβ = (b ),

8 n-points Inequlities o Hermite-Hdmrd Tpe 45 Moreover, we hve ( ) h ddβ = Utilising (8), we get the desired result (6). ( ) β h ddβ. Remrk Let : C X C be convex unction on the convex subset C o rel or complex liner spce X. Then or n x, C nd b > we hve ( ) x + (9) (b ) { [( t) x + t] dt [ b b (b ) () + (x). [ ] [ x + (β + ) (β + ) x + + () () ( βx + ) ddβ + ] () + (x) ]} ddβ The second nd third inequlities re obvious rom (6) or h (t) = t. B the convexit o we hve { [ ] [ ]} x + (β + ) (β + ) x + + () () [ {[ ] [ ]}] x + (β + ) (β + ) x + + () () ( ) x + = or n, β with >. I we multipl this inequlit b [, b] we get +β nd integrte on the squre { [ ] [ x + (β + ) (β + ) x + + () () ( ) x + b b ( ) x + ddβ = (b ), ]} ddβ

9 46 S. S. Drgomir since we know tht This proves the irst inequlit in (9). B the convexit o we lso hve ( ) βx + ddβ = (b ). β (x) + () or n, β with >. Integrting on the squre [, b] we get (x) ( βx + ) ddβ β ddβ + () = (b ) [ () + (x)], which proves the lst inequlit in (9). ddβ Let (X, ) be normed liner spce over the rel or complex number ields. Then or n x, X, p nd b > we hve: p x + () (b ) ( t) x + t p dt [ b b (b ) p + x p. { x + (β + ) () βx + p + (β + ) x + () p ddβ + p + x p ] The cse o Breckner s-convexit is s ollows: p} ddβ Remrk 3 Assume tht the unction : C X [, ) is Breckner s-convex unction with s (, ). Let, x C with x nd ssume tht

10 n-points Inequlities o Hermite-Hdmrd Tpe 47 the mpping [, ] t [( t) x + t] is Lebesgue integrble on [, ]. Then or n b > we hve s (b ) s + { [( t) x + t] dt [ b b (b ) We lso hve the norm inequlities: s (b ) ( t) x + t s dt [ b b (b ) [ ] [ x + (β + ) (β + ) x + + () () ( ) βx + ddβ + { x + (β + ) () βx + or n x, X, normed liner spce. s ] () + (x). + (β + ) x + () s ddβ + s + x s ], ]} ddβ s} ddβ () () 3 Inequlities or n-points In order to extend the bove results or n-points, we need the ollowing representtion o the integrl tht is o interest in itsel. Theorem 3 Let : C X C be deined on the convex subset C o rel or complex liner spce X. Assume tht or x, C with x the mpping [, ] (( t) x + t) C is Lebesgue integrble on [, ]. Then or n prtition = λ < λ <... < λ n < λ n = with n, we hve the representtion n (( t) x + t) dt = (λ j+ λ j ) j= {( u) [( λ j ) x + λ j ] + u [( λ j+ ) x + λ j+ ]} du. (3)

11 48 S. S. Drgomir Proo. We hve In the integrl λj+ n (( t) x + t) dt = consider the chnge o vrible Then j= λj+ λ j (( t) x + t) dt. (4) λ j (( t) x + t) dt, j {,..., n }, u := λ j+ λ j (t λ j ), t [λ j, λ j+ ]. du = λ j+ λ j dt, u = or t = λ j, u = or t = λ j+, t = ( u) λ j + uλ j+ nd λj+ λ j (( t) x + t) dt (5) = (λ j+ λ j ) [( ( u) λ j uλ j+ ) x + (( u) λ j + uλ j+ ) ] du = (λ j+ λ j ) [( u + u ( u) λ j uλ j+ ) x + (( u) λ j + uλ j+ ) ] du = (λ j+ λ j ) = [(( u) ( λ j ) + u ( λ j+ )) x + (( u) λ j + uλ j+ ) ] du {( u) [( λ j ) x + λ j ] + u [( λ j+ ) x + λ j+ ]} du or n j {,..., n }. Mking use o (4) nd (5) we deduce the desired result (3). The ollowing prticulr cse is o interest nd hs been obtined in [7]. Corollr 3 In the the ssumptions o Theorem 3 we hve (( t) x + t) dt = λ or n λ [, ]. + ( λ) {( u) x + u [( λ) x + λ]} du (6) {( u) [( λ) x + λ] + u} du

12 n-points Inequlities o Hermite-Hdmrd Tpe 49 Proo. Follows rom (3) b choosing = λ λ = λ λ =. The ollowing result holds or h-convex unctions: Theorem 4 Let : C X C be deined on the convex subset C o rel or complex liner spce X nd is h-convex on C with h L [, ]. Assume tht or x, C with x the mpping [, ] (( t) x + t) R is Lebesgue integrble on [, ]. Then or n prtition = λ < λ <... < λ n < λ n = with n, we hve the inequlities h ( n ) (λ j+ λ j ) j= n (( t) x + t) dt {( λ ) j + λ j+ x + λ } j + λ j+ (λ j+ λ j ) [ (( λ j ) x + λ j ) + (( λ j+ ) x + λ j+ )] j= h (u) du. (7) Proo. Since is h-convex, then {( u) [( λ j ) x + λ j ] + u [( λ j+ ) x + λ j+ ]} h ( u) (( λ j ) x + λ j ) + h (u) (( λ j+ ) x + λ j+ ) or n u [, ] nd or n j {,..., n }. Integrting this inequlit over u [, ] we get {( u) [( λ j ) x + λ j ] + u [( λ j+ ) x + λ j+ ]} du {h ( u) (( λ j ) x + λ j ) + h (u) (( λ j+ ) x + λ j+ )} du = (( λ j ) x + λ j ) h ( u) du + (( λ j+ ) x + λ j+ ) = [ (( λ j ) x + λ j ) + (( λ j+ ) x + λ j+ )] h (u) du, h (u) du or n j {,..., n }. Multipling this inequlit b λ j+ λ j nd summing over j rom to n we get, vi the equlit (3), the second inequlit in (7).

13 5 S. S. Drgomir Since is h-convex, then or n v, w C we lso hve I we write this inequlit or (v) + (w) ( ) v + w h ( ). v = ( u) [( λ j ) x + λ j ] + u [( λ j+ ) x + λ j+ ] nd w = u [( λ j ) x + λ j ] + ( u) [( λ j+ ) x + λ j+ ] nd tke into ccount tht v + w = {[( λ j) x + λ j ] + [( λ j+ ) x + λ j+ ]} ( = λ ) j + λ j+ x + λ j + λ j+, then we get {( u) [( λ j ) x + λ j ] + u [( λ j+ ) x + λ j+ ]} (8) + {u [( λ j ) x + λ j ] + ( u) [( λ j+ ) x + λ j+ ]} {( h ( ) λ ) j + λ j+ x + λ } j + λ j+ or n u [, ] nd j {,..., n }. Integrting the inequlit (8) over u [, ] we get {( u) [( λ j ) x + λ j ] + u [( λ j+ ) x + λ j+ ]} du (9) + {u [( λ j ) x + λ j ] + ( u) [( λ j+ ) x + λ j+ ]} du {( h ( ) λ ) j + λ j+ x + λ } j + λ j+ or n j {,..., n }. Since = {( u) [( λ j ) x + λ j ] + u [( λ j+ ) x + λ j+ ]} du {u [( λ j ) x + λ j ] + ( u) [( λ j+ ) x + λ j+ ]} du,

14 n-points Inequlities o Hermite-Hdmrd Tpe 5 then b (9) we get {( u) [( λ j ) x + λ j ] + u [( λ j+ ) x + λ j+ ]} du {( h ( ) λ ) j + λ j+ x + λ } j + λ j+ or n j {,..., n }. Multipling this inequlit b λ j+ λ j nd summing over j rom to n we get, vi the equlit (3), the irst inequlit in (7). Remrk 4 I we tke in (7) = λ λ = λ λ =, then we get the irst two inequlities in (4). The cse o convex unctions is s ollows: Corollr 4 Let : C X R be convex unction on the convex subset C o rel or complex liner spce X. Then or n prtition = λ < λ <... < λ n < λ n = with n, nd or n x, C we hve the inequlities ( ) x + n (λ j+ λ j ) j= n (( t) x + t) dt {( λ ) j + λ j+ x + λ } j + λ j+ (λ j+ λ j ) [ (( λ j ) x + λ j ) + (( λ j+ ) x + λ j+ )] j= (x) + (). (3) Proo. The second nd third inequlities in (3) ollows rom (7) b tking h (t) = t. B the Jensen discrete inequlit ( m m ) p j (z j ) p j z j, j= j=

15 5 S. S. Drgomir where p j, j {,..., m} with m j= p j = nd z j C, j {,..., m} we hve n (λ j+ λ j ) j= = = { n j= {( n {( ) x + } {( λ ) j + λ j+ x + λ } j + λ j+ [( (λ j+ λ j ) λ ) j + λ j+ x + λ j + λ j+ n ( ) j= λ j+ λj) (λ j+ λ j ) x + j= ( ) x + = ] } n ( } j= λ j+ λj) nd the irst prt o (3) is proved. B the convexit o we lso hve n (λ j+ λ j ) [ (( λ j ) x + λ j ) + (( λ j+ ) x + λ j+ )] j= n (λ j+ λ j ) [( λ j ) (x) + λ j () + ( λ j+ ) (x) + λ j+ ()] j= n = (λ j+ λ j ) [( (λ j + λ j+ )) (x) + (λ j + λ j+ ) ()] j= ( n n = (λ j+ λ j ) j= = (x) + (), j= ( ) ) n λ j+ λ j (x) + j= ( ) λ j+ λ j () which proves the lst prt o (3). Remrk 5 Let (X, ) be normed liner spce over the rel or complex number ields. Then or n prtition = λ < λ <... < λ n < λ n = with n,

16 n-points Inequlities o Hermite-Hdmrd Tpe 53 nd or n x, X we hve the inequlities p x + n ( (λ j+ λ j ) λ ) j + λ j+ x + λ j + λ j+ j= n ( t) x + t p dt (λ j+ λ j ) [ ( λ j ) x + λ j p + ( λ j+ ) x + λ j+ p ] j= x p + p, where p. p (3) Corollr 5 Let : C X R be deined on convex subset C o rel or complex liner spce X nd is Breckner s-convex on C with s (, ). Assume tht or x, C with x the mpping [, ] (( t) x + t) R is Lebesgue integrble on [, ]. Then or n prtition we hve the inequlities s n (λ j+ λ j ) j= s + = λ < λ <... < λ n < λ n = with n, (( t) x + t) dt {( λ ) j + λ j+ x + λ } j + λ j+ n (λ j+ λ j ) [ (( λ j ) x + λ j ) + (( λ j+ ) x + λ j+ )]. j= (3) Since, or s (, ), the unction (x) = x s is Breckner s-convex on the normed liner spce X, then b (3) we get or n x, X s n ( (λ j+ λ j ) j= s + ( t) x + t s dt λ j + λ j+ ) x + λ j + λ j+ n (λ j+ λ j ) [ ( λ j ) x + λ j s + ( λ j+ ) x + λ j+ s ]. j= s (33)

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18 n-points Inequlities o Hermite-Hdmrd Tpe 55 [] G. Cristescu, Hdmrd tpe inequlities or convolution o h-convex unctions. Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexit 8 (), 3. [3] S. S. Drgomir, An inequlit improving the irst Hermite-Hdmrd inequlit or convex unctions deined on liner spces nd pplictions or semi-inner products. J. Inequl. Pure Appl. Mth. 3 (), no., Article 3, 8 pp. [4] S. S. Drgomir, An inequlit improving the irst Hermite-Hdmrd inequlit or convex unctions deined on liner spces nd pplictions or semi-inner products, J. Inequl. Pure Appl. Mth. 3 (), No., Article 3. [5] S. S. Drgomir, An inequlit improving the second Hermite-Hdmrd inequlit or convex unctions deined on liner spces nd pplictions or semi-inner products, J. Inequl. Pure Appl. Mth. 3 (), No.3, Article 35. [6] S. S. Drgomir, Opertor Inequlities o Ostrowski nd Trpezoidl Tpe. Springer Bries in Mthemtics. Springer, New York,. x+ pp. ISBN: [7] S. S. Drgomir, Inequlities o Hermite-Hdmrd tpe or h-convex unctions on liner spces, Preprint RGMIA Res. Rep. Coll. 6 (3), Art. 7 [Online [8] S. S. Drgomir nd S. Fitzptrick, The Hdmrd inequlities or s- convex unctions in the second sense. Demonstrtio Mth. 3 (999), no. 4, [9] S. S. Drgomir nd S. Fitzptrick,The Jensen inequlit or s-breckner convex unctions in liner spces. Demonstrtio Mth. 33 (), no., [] S. S. Drgomir nd B. Mond, On Hdmrd s inequlit or clss o unctions o Godunov nd Levin. Indin J. Mth. 39 (997), no., 9. [] S. S. Drgomir nd C. E. M. Perce, On Jensen s inequlit or clss o unctions o Godunov nd Levin. Period. Mth. Hungr. 33 (996), no., 93. [] S. S. Drgomir nd C. E. M. Perce, Qusi-convex unctions nd Hdmrd s inequlit, Bull. Austrl. Mth. Soc. 57 (998),

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20 n-points Inequlities o Hermite-Hdmrd Tpe 57 [35] J. Pečrić nd S. S. Drgomir, A generliztion o Hdmrd s inequlit or isotonic liner unctionls, Rdovi Mt. (Srjevo) 7 (99), 3 7. [36] M. Rdulescu, S. Rdulescu nd P. Alexndrescu, On the Godunov- Levin-Schur clss o unctions. Mth. Inequl. Appl. (9), no. 4, [37] M. Z. Srik, A. Sglm, nd H. Yildirim, On some Hdmrd-tpe inequlities or h-convex unctions. J. Mth. Inequl. (8), no. 3, [38] E. Set, M. E. Özdemir nd M. Z. Srık, New inequlities o Ostrowski s tpe or s-convex unctions in the second sense with pplictions. Fct Univ. Ser. Mth. Inorm. 7 (), no., [39] M. Z. Srik, E. Set nd M. E. Özdemir, On some new inequlities o Hdmrd tpe involving h-convex unctions. Act Mth. Univ. Comenin. (N.S.) 79 (), no., [4] M. Tunç, Ostrowski-tpe inequlities vi h-convex unctions with pplictions to specil mens. J. Inequl. Appl. 3, 3:36. [4] S. Vrošnec, On h-convexit. J. Mth. Anl. Appl. 36 (7), no., S. S. Drgomir Mthemtics, College o Engineering & Science Victori Universit, PO Box 448 Melbourne Cit, MC 8, Austrli. School o Computer Science & Applied Mthemtics, Universit o the Witwtersrnd, Privte Bg 3, Johnnesburg 5, South Aric sever.drgomir@vu.edu.u Plese, cite to this pper s published in Armen. J. Mth., V. 8, N. (6), pp