The inequality (1.2) is called Schlömilch s Inequality in literature as given in [9, p. 26]. k=1

Transcription

1 THE TEACHING OF MATHEMATICS 2018, Vol XXI, 1, pp HYBRIDIZATION OF CLASSICAL INEQUALITIES WITH EQUIVALENT DYNAMIC INEQUALITIES ON TIME SCALE CALCULUS Muhmmd Jibril Shhb Shir Abstrct The im of this pper is to present some clssicl inequlities nd to form their symmetric dynmic versions on time scle clculus We lso present tht their symmetric dynmic versions on time scles re equivlent MthEduc Subject Clssifiction: H35 MSC Subject Clssifiction: 97H30, 26D15 Key words nd phrses: Bernoulli s inequlity; Rdon s inequlity; Rogers-Hölder s inequlity; Schlömilch s inequlity; the weighted power men inequlity; time scles 1 Introduction We introduce here some well known clssicl inequlities The weighted power men inequlity given in [6, Theorem 105], [9, pp 12 15] nd [14] is defined s follows If x 1, x 2,, x n re nonnegtive rel numbers nd p 1, p 2,, p n re positive rel numbers, then for > 0, we hve 11) p1 x η1 1 + p 2x η p nx n p 1 + p p n p1 x η2 1 + p 2x η p nx n p 1 + p p n The inequlity 12) is clled Schlömilch s Inequlity in literture s given in [9, p 26] 12) If x 1, x 2,, x n re nonnegtive rel numbers nd 0 < <, then 1 n n x k unless the x k for k N re ll equl < 1 n n x k, Rogers-Hölder s Inequlity, which is commonly known s Hölder s Inequlity, is n importnt nd well known inequlity This inequlity hs mny pplictions History of Rogers-Hölder s Inequlity is given in [13] Rogers-Hölder s Inequlity ws first found by Rogers in 1888 nd then by Hölder in 1889 s given in [11]

2 Clssicl inequlities with equivlent inequlities on time scles 39 Jcob Bernoulli in 1689 proved the clssicl Bernoulli s Inequlity s given in [9], [13] Rdon s Inequlity is widely studied by mny uthors s it hs mny pplictions s given in [16] Rdon s inequlity in generlized form is given in [8] The upcoming theorem shows tht Rogers-Hölder s Inequlity, Bernoulli s Inequlity, Rdon s Inequlity nd generlized Rdon s Inequlity re equivlent s given in [8] Theorem 11 The following inequlities re equivlent: 1) Rogers-Hölder s Inequlity If p > 1, 1 p + 1 q = 1, where x k nd y k for ll k = 1, 2,, n re positive rels, then the discrete version of Rogers-Hölder s Inequlity is n n p n q 13) x k y k 2) Bernoulli s Inequlity x p k If ϕx) 0, x [, b], where p > 1, then 14) ϕ p x) 1 + pϕx) 1) If ϕx) 0, x [, b], where 0 < ξ < 1, then the reversed version of Bernoulli s Inequlity is 15) ϕ ξ x) 1 + ξϕx) 1) 3) Rdon s Inequlity If n N, x k 0 nd y k > 0, k {1, 2,, n} nd γ 0, then n ) γ+1 x k n x γ+1 k 16) n ) γ y γ y k k 4) Generlized Rdon s Inequlity If n N, x k 0 nd y k > 0, k {1, 2,, n}, γ 0 nd ζ 1, then n ) γ+ζ x k y ζ 1 k n x γ+ζ k 17) n ) γ+ζ 1 y γ, k with equlity if nd only if x1 y 1 y ζ k = x2 y 2 = = xn y n We will prove these results on time scle clculus The theory of time scle clculus is pplied to revel the symmetry of continuous nd discrete nd to combine them in one comprehensive form In time scle clculus, results re unified y q k

3 40 M J S Shir nd extended This hybrid theory is lso widely pplied to dynmic inequlities Reserch work on dynmic inequlities ws done by R Agrwl, G Anstssiou, M Bohner, A Peterson, D O Regn, S Sker nd mny other uthors In this pper, it is ssumed tht ll considerble integrls exist nd re finite nd T is time scle,, b T with < b nd n intervl [, b] T mens the intersection of the rel intervl with the given time scle 2 Preliminries Time scle clculus ws initited by S Hilger s given in [10] It is studied s delt clculus, nbl clculus nd dimond-α clculus A time scle is n rbitrry nonempty closed subset of the set of rel numbers We need here bsic concepts of delt clculus The results of delt clculus re dpted from [4, 5] For t T, forwrd jump opertor σ : T T is defined by σt) := inf{s T : s > t} The mpping µ : T R + 0 = [0, ) such tht µt) := σt) t is clled the forwrd grininess function The bckwrd jump opertor ρ : T T is defined by ρt) := sup{s T : s < t} The mpping ν : T R + 0 = [0, ) such tht νt) := t ρt) is clled the bckwrd grininess function If σt) > t, we sy tht t is right-scttered, while if ρt) < t, we sy tht t is left-scttered Also, if t < sup T nd σt) = t, then t is clled right-dense, nd if t > inf T nd ρt) = t, then t is clled left-dense If T hs left-scttered mximum M, then T k = T {M}, otherwise T k = T For function f : T R, the delt derivtive f is defined s follows Let t T k If there exists f t) R such tht for ll ɛ > 0, there exists neighborhood U of t, such tht fσt)) fs) f t)σt) s) ɛ σt) s, for ll s U, then f is sid to be delt differentible t t, nd f t) is clled the delt derivtive of f t t A function f : T R is sid to be right-dense continuous rd-continuous), if it is continuous t ech right-dense point nd there exists finite left limit t every left-dense point The set of ll rd-continuous functions is denoted by C rd T, R) Definition 21 [4, 5] A function F : T R is clled delt ntiderivtive of f : T R, provided tht F t) = ft) holds for ll t T k Then the delt integrl of f is defined by ft) t = F b) F )

4 Clssicl inequlities with equivlent inequlities on time scles 41 The following results of nbl clculus re tken from [3, 4, 5] If T hs right-scttered minimum m, then T k = T {m}, otherwise T k = T For f : T R, function f is clled nbl differentible t t T k, with nbl derivtive f t), if there exists f t) R such tht for ny given ɛ > 0, there exists neighborhood V of t, such tht for ll s V fρt)) fs) f t)ρt) s) ɛ ρt) s, A function f : T R is sid to be left-dense continuous ld-continuous), provided it is continuous t ll left-dense points in T nd its right-sided limits exist finite) t ll right-dense points in T The set of ll ld-continuous functions is denoted by C ld T, R) Definition 22 [3, 4, 5] A function G : T R is clled the nbl ntiderivtive of g : T R, provided tht G t) = gt) holds for ll t T k Then the nbl integrl of g is defined by gt) t = Gb) G) Now we present short introduction of dimond-α derivtive s given in [1, 19] Let T be time scle nd ft) be differentible on T in the nd senses For t T k k, where Tk k = Tk T k, dimond-α dynmic derivtive f α t) is defined by f α t) = αf t) + 1 α)f t), 0 α 1 Thus, f is dimond-α differentible if nd only if f is nd differentible The dimond-α derivtive reduces to the stndrd -derivtive for α = 1, or the stndrd -derivtive for α = 0 It represents weighted dynmic derivtive for α 0, 1) Theorem 23 [19] Let f, g : T R be dimond-α differentible t t T Then i) f ± g : T R is dimond-α differentible t t T, with f ± g) α t) = f α t) ± g α t) ii) fg : T R is dimond-α differentible t t T, with fg) α t) = f α t)gt) + αf σ t)g t) + 1 α)f ρ t)g t) iii) For gt)g σ t)g ρ t) 0, f g : T R is dimond-α differentible t t T, with ) α f t) = f α t)g σ t)g ρ t) αf σ t)g ρ t)g t) 1 α)f ρ t)g σ t)g t) g gt)g σ t)g ρ t)

5 42 M J S Shir Definition 24 [19] Let, t T nd h : T R Then the dimond-α integrl from to t of h is defined by t hs) α s = α t hs) s + 1 α) provided tht there exist delt nd nbl integrls of h on T t hs) s, 0 α 1, Theorem 25 [19] Let, b, t T, c R Assume tht fs) nd gs) re α -integrble functions on [, b] T Then i) t [fs) ± gs)] α s = t fs) α s ± t gs) α s; ii) t cfs) α s = c t fs) α s; iii) t fs) α s = t fs) α s; iv) t fs) α s = fs) α s + t b fs) α s; v) fs) α s = 0 To proceed further, we need the following result Theorem 26 [1] Let, b T nd c, d R Suppose tht g C[, b] T, c, d)) nd h C[, b] T, R) with hs) α s > 0 If F Cc, d), R) is convex, then the generlized Jensen s Inequlity is 21) F hs) gs) ) α s hs) hs) F gs)) α s α s hs) α s If F is strictly convex, then the inequlity cn be replced by < Definition 27 [7] A function f : T R is clled convex on I T = I T, where I is n intervl of R open or closed), if 22) fλt + 1 λ)s) λft) + 1 λ)fs), for ll t, s I T nd ll λ [0, 1] such tht λt + 1 λ)s I T The function f is strictly convex on I T if the inequlity 22) is strict for distinct t, s I T nd λ 0, 1) The function f is concve respectively, strictly concve) on I T, if f is convex respectively, strictly convex) 3 Min results In order to present our min results, we first present n extension of generlized Rdon s inequlity by pplying Bernoulli s Inequlity vi time scles Theorem 31 wx), gx) 0 Let w, f, g C[, b] T, R) be α -integrble functions, where

6 Clssicl inequlities with equivlent inequlities on time scles 43 i) If γ 0 nd ζ 1, then ) b γ+ζ wx) fx) gx) ζ 1 α x 31) ) b γ+ζ 1 wx) gx) ζ α x ii) If 0 < γ + ζ < 1, then ) b γ+ζ wx) fx) gx) ζ 1 α x 32) ) b γ+ζ 1 wx) gx) ζ α x wx) fx) γ+ζ gx) γ α x wx) fx) γ+ζ gx) γ α x Equlity occurs in 31) nd 32) if nd only if fx) = cgx), where c is rel constnt Proof We prove this result by pplying Bernoulli s Inequlity The inequlity 31) cn be rerrnged s ) b γ+ζ 1 b wx) gx) ζ α x wx) fx) γ+ζ gx) γ α x wx) fx) gx) ζ 1 α x Let us consider ) b γ+ζ 1 wx) fx) γ+ζ wx) gx) ζ α x gx) γ ) b γ+ζ wx) fx) gx) ζ 1 α x wx) gx) ζ fx) = wx) gx) ζ α x b wx) gx) ζ α x gx) wx) fx) gx) ζ 1 α x wx) gx) ζ = wx) gx) ζ α x = 1 + fx) wx) gx) ζ α x gx) wx) gx) ζ wx) gx) ζ α x gx) wx) fx) gx) ζ 1 α x 1 + γ + ζ) fx) wx) gx) ζ α x gx) wx) gx) ζ wx) gx) ζ α x { + γ + ζ) wx) fx) gx) ζ 1 wx) fx) gx) ζ 1 α x ) γ+ζ 1 ) γ+ζ wx) fx) gx) ζ 1 α x gx) wx) fx) gx) ζ 1 α x ) γ+ζ ) wx) fx) gx) ζ 1 α x } wx) gx) ζ wx) gx) ζ α x

7 44 M J S Shir Then, integrting from to b, we get wx) fx) γ+ζ gx) γ α x ) b γ+ζ 1 wx) gx) ζ α x ) b γ+ζ 1 + γ + ζ)1 1) wx) fx) gx) ζ 1 α x We get the required result given in 31) The inequlity given in 31) is reversed if we set 0 < γ + ζ < 1 It is cler tht equlity holds in 31) nd 32), if nd only if fx) = cgx), where c is rel constnt number In the following corollry, we give dynmic Rdon s Inequlity on time scles, which is ctully the reduced form of generlized Rdon s Inequlity Corollry 32 Let w, f, g C[, b] T, R) be α -integrble functions, where wx), gx) 0 If γ 0, then 33) ) b γ+1 wx) fx) α x ) b γ wx) gx) α x wx) fx) γ+1 gx) γ α x Proof For ζ = 1 31) reduces to 33) Next we prove tht generlized Rdon s Inequlity given in 31) nd Rdon s Inequlity given in 33) re equivlent Theorem 33 The following inequlities re equivlent on dynmic time scle clculus: 1) Generlized Rdon s Inequlity, 2) Rdon s Inequlity Proof It is cler from Corollry 32 tht generlized Rdon s Inequlity implies Rdon s Inequlity To prove the converse, replce γ by γ + ζ 1, where ζ 1 nd lso replce wx) by wx) gx) ζ 1 for g C[, b] T, R {0}) in 33) nd we get 31) The upcoming corollry gives us the weighted power men inequlity on time scles Corollry 34 Let w, f C[, b] T, R) be α -integrble functions, where wx) 0 If > 0, then 34) wx) fx) η1 α x wx) α x wx) fx) η2 α x wx) α x

8 Clssicl inequlities with equivlent inequlities on time scles 45 Proof Set γ 0, ζ = 1, 1 + γ = 1 nd gx) = 1 Then 31) becomes 35) wx) fx) α x wx) α x ) ) 1 wx) fx) α x Dividing 35) by wx) α x nd replcing fx) by fx) η1, then tking power > 0, we get our clim 1 The following theorem shows tht the generlized Rdon s Inequlity given in 31) nd the weighted power men inequlity given in 34) re equivlent Theorem 35 The following inequlities re equivlent on dynmic time scle clculus: 1) Generlized Rdon s Inequlity, 2) The weighted power men inequlity Proof It is cler from Corollry 34 tht generlized Rdon s Inequlity implies the weighted power men inequlity In order to prove the converse, replce fx) by fx) 1, nd tke power > 0 nd = γ + ζ 1, where γ 0 nd ζ 1 Then 34) tkes the form 36) wx) fx) ) γ+ζ α x wx) wx) fx) γ+ζ α x α x wx) α x Now we replce wx) by wx) gx) ζ C[, b] T, R {0}) in 36) Then we get nd lso replce fx) by fx) gx) for g 37) ) γ+ζ wx) fx) gx) ζ 1 α x wx) gx) ζ α x wx) fx) γ+ζ gx) γ α x wx) gx) ζ α x Multiplying 37) by wx) gx) ζ α x, we get the required clim Remrk 36 If we set α = 1, T = Z, nd wx) is replced by p k for k {1, 2,, n}, where {p k } is set of positive rel numbers nd fx) is replced by x k for k {1, 2,, n}, where {x k } is set of nonnegtive rel numbers, then 34) reduces to 11) Further, if we set wx) = 1, then 34) reduces to 12) Now we give Schlömilch s Inequlity on time scles Corollry 37 Let w, f C[, b] T, R) be α -integrble functions, where wx) 0 If > 0, then 38) wx) fx) η 1 α x wx) fx) η 2 α x

9 46 M J S Shir Proof If we set wx) α x = 1 in 34), then we get 38) Next we show tht weighted power men inequlity given in 34) is equivlent to Schlömilch s Inequlity given in 38) Theorem 38 The following inequlities re equivlent on dynmic time scle clculus: 1) The weighted power men inequlity, 2) Schlömilch s Inequlity Proof It is cler from Corollry 37 tht the weighted power men inequlity wx) implies Schlömilch s Inequlity Replcing wx) by we obtin 34) wx) αx, from 38) Remrk 39 Let w, f C rd [, b] T, [0, )) Then for α = 1, the inequlity given in 38) tkes the form 39) wx)f η 1 x) x wx)f η 2 x) x, s given in [12, Lemm A] Let w, f C[, b] T, [0, )) Then 38) tkes the form 310) wx)f η 1 x) α x wx)f η 2 x) α x, s given in [20, Lemm 34] Now we give nother form of Schlömilch s Inequlity on time scles Corollry 310 Let f C[, b] T, R) be α -integrble function If > 0, then 311) fx) α x b fx) α x b Proof Set wx) = 1 in 34), nd we get 311) Exmple 311 Set = 1 Then 311) tkes the form 312) ) η2 1 1 b η2 fx) α x) b fx) α x In the upcoming result, using generlized Jensen s Inequlity, we find Rogers- Hölder s Inequlity on time scles Rogers-Hölder s Inequlity is lso given in [1, 15]

10 Clssicl inequlities with equivlent inequlities on time scles 47 1 p + 1 q Corollry 312 Let w, f, g C[, b] T, R) be α -integrble functions nd = 1 with p > 1 Then 313) wx) fx)gx) α x wx) fx) p p b α x wx) gx) q q α x Proof The inequlity given in 31) cn be rerrnged s wx) gx) ζ ) 314) fx) γ+ζ b wx) gx) ζ α x gx) α x b wx) gx) ζ fx) gx) γ+ζ α x wx) gx) ζ α x Let γ > 0, ζ = 1 nd set γ + 1 = p > 1 The function F : [0, ) R defined by F x) = x γ+1 is convex for x [0, ), nd 314) is similr to generlized Jensen s Inequlity given in 21) Then 314) tkes the form 315) wx) fx) α x ) p ) p 1 ) wx) gx) α x wx) fx) p gx) 1 p α x Further, we replce wx) by wx) gx) q p nd fx) by fx) gx) 1 q p Tking the power 1 p > 0, 315) reduces to 313) Using the previously obtined results, we conclude tht severl dynmic inequlities re equivlent Theorem 313 The following inequlities re equivlent on time scle clculus: 1) Generlized Rdon s Inequlity, 2) Rdon s Inequlity, 3) The weighted power men inequlity, 4) Schlömilch s Inequlity, 5) Rogers-Hölder s Inequlity, 6) Bernoulli s Inequlity Proof It follows from Theorems 31, 33, 35, 38 nd Corollry 312 tht 6) = 1), the inequlities 1), 2), 3) nd 4) re equivlent, nd tht 1) = 5) It remins to prove tht 5) = 6)

11 48 M J S Shir Without loss of generlity, we my suppose tht wx) fx) p p α x wx) gx) q α x Then Rogers-Hölder s Inequlity given in 313) tkes the form wx) fx) wx) fx) p α x p q gx) wx) gx) q α x 0 q α x 1 = 1 p + 1 q = 1 wx) fx) p α x p b wx) fx) p α x + 1 wx) gx) q α x q b wx) gx) q α x So, 316) fx) wx) fx) p α x p gx) wx) gx) q q α x 1 fx) p p b wx) fx) p α x + 1 gx) q q b wx) gx) q α x Let ux) = cn be written s fx) wx) fx) p α x p nd vx) = gx) wx) gx) q Then 316) α x q 317) ux)vx) 1 p up x) + 1 q vq x), where ux) nd vx) re nonnegtive rel functions Tking 1 q = 1 1 p, where 1 p = ξ < 1, 317) cn be written s 318) u ξ x)v 1 ξ x) ξux) + 1 ξ)vx) Dividing 318) by vx) nd tking the substitution ϕx) = ux) vx), we get ϕ ξ x) ξϕx) + 1 ξ), which is the required Bernoulli s Inequlity Remrk 314 If we set α = 1, T = Z, wx) = 1, fx) = x k nd gx) = y k for k {1, 2,, n}, n N, where x k nd y k re positive rels, then 313) reduces to 13) If we set α = 1, T = Z, wx) = 1, fx) = x k nd gx) = y k for k {1, 2,, n}, n N, where x k nd y k re nonnegtive rels nd y k 0, then the discrete version of 31) reduces to 17) for γ 0 nd ζ 1 If ζ = 1, then the discrete version of 17) reduces to 16) for γ 0

12 Clssicl inequlities with equivlent inequlities on time scles 49 It is cler from Theorem 313 tht Rogers-Hölder s Inequlity, Bernoulli s Inequlity, Generlized Rdon s Inequlity nd Rdon s Inequlity re lso equivlent So Theorem 11 is just prt of Theorem 313 In the following exmple, we present some dynmic inequlities such s Rdon s Inequlity, the weighted power men inequlity nd Schlömilch s Inequlity on quntum clculus Exmple 315 If we set [, b] T = [q m, q n ] q N 0 for q > 1 nd m < n, where m, n N nd N 0 is the set of nonnegtive integers, then q n n 1 fx) α x = q 1) q i [αfq i ) + 1 α)fq i+1 )] q m If γ 0, then 33) tkes the form [ n 1 [ n 1 q { i α wq i ) fq i ) + 1 α) wq i+1 ) fq i+1 ) }] γ+1 ] γ q i {α wq i ) gq i ) + 1 α) wq i+1 ) gq i+1 ) } n 1 If > 0, then 34) tkes the form n 1 { q i α wqi ) fq i ) γ+1 gq i ) γ + 1 α) wqi+1 ) fq i+1 ) γ+1 } gq i+1 ) γ q i {α wq i ) fq i ) + 1 α) wq i+1 ) fq i+1 ) } n 1 q i {α wq i ) + 1 α) wq i+1 ) } n 1 q i {α wq i ) fq i ) η2 + 1 α) wq i+1 ) fq i+1 ) η2 } n 1 q i {α wq i ) + 1 α) wq i+1 ) } If > 0, then 38) tkes the form [ n 1 q 1) q i { α wq i ) fq i ) η1 + 1 α) wq i+1 ) fq i+1 ) η1}] 1 [ n 1 q 1) q i { α wq i ) fq i ) + 1 α) wq i+1 ) fq i+1 ) }] 1 Finlly, we present integrl dynmic inequlities in two dimensions Dimondα integrl for function of two vribles is defined in [1] 1 1

13 50 M J S Shir Theorem 316 equivlent: 1) Generlized Rdon s Inequlity The following dynmic inequlities in two dimensions re Let wx 1, x 2 ), fx 1, x 2 ), gx 1, x 2 ) C[ i, b i ] 2 T, R) i = 1, 2) be α-integrble functions, where wx 1, x 2 ), gx 1, x 2 ) 0 If γ 0 nd ζ 1, then 1 ) b2 γ+ζ 1 2 wx 1, x 2 ) fx 1, x 2 ) gx 1, x 2 ) ζ 1 α x 1 α x 2 1 ) b2 γ+ζ 1 wx 1, x 2 ) gx 1, x 2 ) ζ α x 1 α x ) Rdon s Inequlity wx 1, x 2 ) fx 1, x 2 ) γ+ζ gx 1, x 2 ) γ α x 1 α x 2 Let wx 1, x 2 ), fx 1, x 2 ), gx 1, x 2 ) C[ i, b i ] 2 T, R) i = 1, 2) be α-integrble functions, where wx 1, x 2 ), gx 1, x 2 ) 0 If γ 0, then 1 ) b2 γ wx 1, x 2 ) fx 1, x 2 ) α x 1 α x 2 1 ) b2 γ wx 1, x 2 ) gx 1, x 2 ) α x 1 α x ) The weighted power men inequlity 1 2 wx 1, x 2 ) fx 1, x 2 ) γ+1 gx 1, x 2 ) γ α x 1 α x 2 Let wx 1, x 2 ), fx 1, x 2 ) C[ i, b i ] 2 T, R) i = 1, 2) be α-integrble functions, where wx 1, x 2 ) 0 If > 0, then wx 1, x 2 ) fx 1, x 2 ) η1 α x 1 α x wx 1, x 2 ) α x 1 α x 2 1 4) Schlömilch s Inequlity wx 1, x 2 ) fx 1, x 2 ) α x 1 α x wx 1, x 2 ) α x 1 α x 2 Let wx 1, x 2 ), fx 1, x 2 ) C[ i, b i ] 2 T, R) i = 1, 2) be α-integrble functions, where wx 1, x 2 ) α x 1 α x 2 = 1 nd wx 1, x 2 ) 0 If > 0 Then wx 1, x 2 ) fx 1, x 2 ) α x 1 α x wx 1, x 2 ) fx 1, x 2 ) η2 α x 1 α x 2

14 Clssicl inequlities with equivlent inequlities on time scles 51 5) Rogers-Hölder s Inequlity Let wx 1, x 2 ), fx 1, x 2 ), gx 1, x 2 ) C[ i, b i ] 2 T, R) i = 1, 2) be α-integrble functions nd 1 p + 1 q = 1 with p > 1 Then wx 1, x 2 ) fx 1, x 2 )gx 1, x 2 ) α x 1 α x wx 1, x 2 ) fx 1, x 2 ) p α x 1 α x p 2 wx 1, x 2 ) gx 1, x 2 ) q α x 1 α x 2 q 6) Bernoulli s Inequlity If ϕx 1, x 2 ) C[ i, b i ] 2 T, [0, )) i = 1, 2), where p > 1, then ϕ p x 1, x 2 ) 1 + pϕx 1, x 2 ) 1) If ϕx 1, x 2 ) C[ i, b i ] 2 T, [0, )) i = 1, 2), where 0 < ξ < 1, then the reversed version of Bernoulli s Inequlity is ϕ ξ x 1, x 2 ) 1 + ξϕx 1, x 2 ) 1) Proof Similr to the proof of Theorem Conclusion nd future work In this reserch rticle, we hve generlized nd extended some clssicl inequlities Our work shows tht mny clssicl inequlities such s generlized Rdon s Inequlity, Rdon s Inequlity, the weighted power men inequlity, Schlömilch s Inequlity, Rogers-Hölder s Inequlity nd Bernoulli s Inequlity re equivlent on dimond-α clculus If we set α = 1, then we get delt versions of dynmic inequlities nd if we set α = 0, then we get nbl versions of dynmic inequlities Also we get discrete versions of dynmic inequlities, if we put T = Z nd we get continuous versions of dynmic inequlities, if we put T = R In future, we cn find more equivlent dynmic inequlities on dimond-α clculus We cn generlize dynmic inequlities using functionl generliztion such s Schlömilch s nd Rogers-Hölder s Inequlities re given in [20] Dynmic inequlities cn lso be generlized in similr fshion s frctionl dynmic inequlities re generlized using convex functions given in [2, 17], Riemnn-Liouville frctionl integrl given in [2, 18] nd frctionl derivtives given in [2] It will be interesting to present dynmic inequlities in three or more dimensions in more generlized form To present these dynmic inequlities on quntum clculus will lso be n interesting work

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