About Some Inequalities for Isotonic Linear Functionals and Applications

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1 Applied Mthemticl Sciences Vol no HIKARI Ltd Aout Some Inequlities fo Isotonic Line Functionls nd Applictions Loedn Ciudiu Deptment of Mthemtics Politehnic Univesity of Timiso P-t. Victoiei No Timiso Romni Copyight c 04 Loedn Ciudiu. This is n open ccess ticle distiuted unde the Cetive Commons Attiution License which pemits unesticted use distiution nd epoduction in ny medium povided the oiginl wok is popely cited. Astct The im of this ppe is to estlish some extensions of Holde s inequlity Minkowski s inequlity nd Qi s inequlity fo isotonic line functionls tking into ccount tht the time scle Cuchy delt Cuchy nl α-dimond multiple Riemnn nd multiple Leesque integls ll e isotonic line functionls. Then sevel pplictions of these esults fo othe pticul isotonic line functionls will e lso otned. Mthemtics Suject Clssifiction: 6D5 Keywods: Holde s inequlity clculus of time scles Minkowski s inequlity. Intoduction In this ppe we dopt the nottions fom the monogph 4] of Bohne nd Peteson. Fo futhe infomtion concening time scles see 4]. The following esults enuncited in this section will e useful elow in ode to estlish the min esults of this ppe nd cn e found in 4] in 5] in 3] nd in 7]. Mny clssicl inequlities e poved in the monogph ] fo the so-clled isotonic line functionls nd using the fct tht the time-scles integl is n isotonic line functionl we cn use these esults to these integls. Also in 8]

2 890 Loedn Ciudiu ppe othe usul exmples of isotonic line functionls tht e nomlised. Theefoe the esults fom this ppe which tke plce fo such functionls cn e ewitten fo these pticul exmples. Moeove s pplictions Guss type inequlity nd some of its efinements which tke plce fo nomlised isotonic line functionls will e tue fo the time scle Cuchy delt Cuchy nl α-dimond multiple Riemnn nd multiple Leesque integls. We need lso to ecll the esults which will e used elow. Lemm. (9] Coolly 3.3) If f is - integle on ) then fo n ity positive nume α the function f α is -integle on ). Lemm. (9] Theoem 3.6) Let f nd g e -integle functions on ). then thei poduct fg is -integle on ). The following definition is given in 3] 7] nd it is necessy to ecll it hee. Definition. Let E e nonempty set nd L e clss of el-vlued functions f : E R hving the following popeties: (L) If f g L nd R then (f g) L. (L L i.e. if f(t) = fo ll t E then f L. An isotonic line functionl is functionl A : L R hving the following popeties: (A) If f g L nd R then A(f g) = A(f) A(g). (A) If f L nd f(t) 0 fo ll t E then A(f) 0. The mpping A is sid to e nomlised if (A3) A() =. Now we will ecll Holde s inequlity fo isotonic line functionls s it ppes in ]. Theoem. (3]) Let E L nd A such tht (L) (L) (A) nd (A) e stisfied. Fo p define q = p. Assume p w f p w g q wfg L. If p > then () A( wfg ) A p ( w f p )A q ( w g q ). Then inequlity is evesed if 0 < p < nd A( w g q ) > 0 nd it is lso evesed if p < 0 nd A( w f p ) > 0. We enuncite Theoem. fom ] in the cse of these functionls when p > nd then the sme kind of theoem Theoem fom ] in the cse of these functionls when 0 < p <. The poof of the esult fom elow is given in 6].

3 Inequlities fo isotonic line functionls 89 Theoem. Let < p < nd let q = p p e its conjugte exponent L stisfy conditions L L nd A stisfy A A on the set E. If f p g q fg f p g q L A( f p ) > 0 A( g q ) > 0 nd if < p then ( A p ( f p )A q ( g q A( f p g q ) A( fg ) p A ( f p )A ( g q ) () A p ( f p )A q ( g q ) q ( A( f p g q ) A ( f p )A ( g q ) while if p < the tems nd exchnge thei positions in the peceeding p q inequlities.. Sudividing of Holde s inequlity fo isotonic line functionls In this section we give some efinements of Holde s inequlty nd Minkowski s inequlity when 0 < < fo isotonic line functionls nd s pplictions the coesponding inequlities fo on time scles Cuchy delt Cuchy nl nd α-dimond integls. These esults e geneliztions of some esults given in ppes s ] ] 5] 3]. Theoem 3. Let 0 < < nd let s = e its conjugte exponent L stisfies conditions L L nd A stisfies A A on the set E. If hk k s h h k s L e nonnegtive functions A(hk) > 0 A(k s ) > 0 A(h ) > 0 nd if < < then A(hk ( A(h k s ) A A (hk)a (h )A s (k s ) (k s ) (3) A(hk) ( ) ( A(h k s ) A (hk)a (k s ) while if 0 < the tems nd exchnge thei positions in the peceeding inequlities. Poof. Suppose < nd tke p =. Becuse < p we cn pply inequlity () to the functions f = h k nd g = k (hw) whee k = otining: A (hk)a (k s ) ( A(h k s ) A (hk)a (k s ) A(h ) A s ((hw) s( ) )

4 89 Loedn Ciudiu A (hk)a (k s ( ) ( A(h k s ). A (hk)a (k s ) Then the inequlity (3) is ovious. As in ] if 0 < then p < nd it is enough to intechnge nd in inequlity (). p q Now we will conside elow some pticul cses of pevious Holde s inequlity when 0 < <. Consequence. (i) Let 0 < < s = functions h w with k = we hve: ( h(x)w(x) x) ( (h(x)w(x)) ( (h(x)w(x))s( ) x s its conjugte the positive nd h w C d ( ] R). Then h (x)k s (x) x ( ( h(x)k(x) x ks (x) x ( h (x) x ( h(x)w(x) x) ( ) k s s (x) x h (x)k s (x) x ( ( h(x)k(x) x ks (x) x if < nd if 0 < then the sme inequlity holds ut with eplced y. (ii) Let 0 < < s = its conjugte the positive functions h w with k = (h(x)w(x)) ( (h(x)w(x))s( ) x s ( h(x)w(x) x) ( nd h w C ld ( ] R). Then we hve: h (x)k s (x) x ( ( h(x)k(x) x ks (x) x ( h (x) x ( h(x)w(x) x) ( ) k s s (x) x h (x)k s (x) x ( ( h(x)k(x) x ks (x) x if < nd if 0 < then the sme inequlity holds ut with eplced y.

5 (iii) Let 0 < < s = Inequlities fo isotonic line functionls 893 ] R e α -integle functions with with k = h w C d ( ] R). Then we hve: ( h(x)w(x) α x) ( its conjugte the positive functions h w : ( h(x)k(x) α x (h(x)w(x)) ( (h(x)w(x))s( ) αx s h (x)k s (x) α x ( ( h (x) α x k s s (x) α x ( h(x)w(x) α x) ( ) ( h(x)k(x) α x ks (x) α x h (x)k s (x) α x ( nd ks (x) α x if < nd if 0 < then the sme inequlity holds ut with eplced y. Theoem 4. Let 0 < < nd L stisfying conditions L L nd A stisfying A A on the set E. Consideing the nonnegtive functions h w with h w h(h w) w(h w) h (h w) w (h w) L A(h ) > 0 A(w ) > 0 A(w (h w) ) > 0 A(h (h w) ) > 0 nd we hve A ((h w) ) A (h ) (4) A (w ) k = ( ) (h w) A s ((h w) s( ) ) ( ) ( A(h k s ) ( A(w k s ) A (hk)a (k s ) A (wk)a (k s ) if < nd if 0 < then the sme inequlity tkes plce ut with eplced y. Poof. When < tking into ccount hypothesis s in ] we notice tht ( ) A ((h w) (h w) ) = A (h w) = A(hk) A(wk) A s ((h w) s( ) )

6 894 Loedn Ciudiu A (h ) A (w ) y the inequlity (3). ( ) ( ) ( A(h k s ) A (hk)a (k s ) ( A(w k s ) A (wk)a (k s ) Some pticul cses of pevious inequlity will e fomulted elow. Consequence. (i) Let 0 < < the positive functions h w with (h(x)w(x)) k = nd h w C d ( ] R). The following inequlity tkes plce: ( ( (h(x)w(x))s( ) x s h (x) x ( w (x) x ( (h(x) w(x)) x ( ) ( ) h (x)k s (x) x ( ( h(x)k(x) x ks (x) x w (x)k s (x) x ( ( w(x)k(x) x ks (x) x if < nd if 0 < then the sme inequlity holds ut with eplced y. (h(x)w(x)) (ii) Let 0 < < the positive functions h w with k = nd h w C ld ( ] R). The following inequlity tkes plce: ( (h(x) w(x)) x ( h (x) x ( w (x) x ( ) ( ) ( (h(x)w(x))s( ) x s h (x)k s (x) x ( ( h(x)k(x) x ks (x) x w (x)k s (x) x ( ( w(x)k(x) x ks (x) x if < nd if 0 < then the sme inequlity holds ut with eplced y.

7 Inequlities fo isotonic line functionls 895 (iii) Let 0 < < the positive functions h w : ] R e α -integle (h(x)w(x)) functions with k =. The following inequlity tkes plce: ( h (x) α x ( w (x) α x ( (h(x)w(x))s( ) αx s ( (h(x) w(x)) α x ( ) ( ) h (x)k s (x) α x ( ( h(x)k(x) α x ks (x) α x w (x)k s (x) α x ( ( w(x)k(x) α x ks (x) α x if < nd if 0 < then the sme inequlity holds ut with eplced y. 3. Some vints of Qi s inequlity fo isotonic line functionls In this section we give sevel vints of some inequlities fom ] in the cse of isotonic line functionls fo p > nd 0 < < using the coesponding Holde s inequlities. Lemm 3. Let E L nd A e such tht L L A A e stisfied. If f g f p p f g g p q g q p L e positive functions then ( ( ) ) p f min{ p A p g ( ) q } ) f p A A (g) g p q g p q ( A f p g p q whee p > o p < 0 while p q =. Ap (f) A p q (g) Theoem 5. Let E L nd A e such tht L L A A e stisfied. If f f p f p L f is positive nd A(f) A () then ( A p ()A(f p A(f p p ) A p (f) p A (f p )A () tkes plce fo p >.

8 896 Loedn Ciudiu Poof. By Lemm 3 nd hypothesis we hve ( A(f p A(f p p ) = p A (f p )A () ( ) ( f p = A A(f p p ) p p A (f p )A () Ap (f) A p () = Ap (f)a(f) A p ()A () Ap (f). A p (). Consequence 3. Let E L nd A e such tht L L A A e stisfied. If f f p f p L f is positive nd in ddition A is nomlised nd A(f) then A(f p p p A(f ) ( A p (f) p A (f p ) tkes plce fo p >. As pplictions next esults pesent some efinements of some inequlities given y Qi nd Yin ] in the cses of delt time-scle integl the Cuchy nl time-scles integls nd the Cuchy α-dimond time scle integls. Remk. (i) Let T. If f C d (T R) is positive -diffeentile on ( ) nd then f p (x) x p ( ) p f(x) x ( ) ( f p p (x) x ( ( f p (x) x p f(x) x] whee p >. (ii) In the cse of the Cuchy nl time-scles integls nd the Cuchy α-dimond time scle integls simily inequlities cn e stted s ove. Lemm 4. Let E L nd A e such tht L L A A e stisfied. If 0 < < s = nd f g f f g g L e positive functions nd s A(f) > 0 A(g) > 0 then ( ) ( f A A (f) ( A(f g ) g s A s (g) A (f)a (g)

9 Inequlities fo isotonic line functionls 897 when < < while if 0 < the tems nd exchnge thei positions in the peceeding inequlities. Poof. We pply Holde s inequlity fom Theoem 3 when f f g f g g L e positive functions otining: s ( ) f A(f) = A g s nd o A ( ( ) ) f A (g A(f) ( ) g s ) s s A (f)a (g) ( ) f g s A(f) A s (g) A ( f g s g s ) A s (g) ( ) A f g ( ) A (f)a (g) < < nd Then we tke the -th powe on oth sides of the inequlities nd hve: ( ) ( ) f A A (f) A f g ( ). g s A s (g) A (f)a (g) Consequence 4. Let T nd 0 < < s = positive then f (x) g s (x) x ( ( f(x) x) ( f(x) x ( s when < < while if 0 < positions in the peceeding inequlities... If f C d(t R) is f (x)g (x) x ( f(x) x g(x) x the tems nd exchnge thei A vint of Theoem 3. fom ] fo isotonic line functionl cn e the following: Theoem 6. Let E L nd A e such tht L L A A e stisfied. If 0 < < s = f f f L f is positive A(f) > 0 nd A(f) A () then ( ) A f A(f ) A (f) ( ) A (f)a ()

10 898 Loedn Ciudiu when < < while if 0 < the tems nd exchnge thei positions in the peceeding inequlities. Poof. By Lemm 4 nd hypothesis we hve ( ) ( ) f A(f f ) = A = A when < <. ( ) ( ) f A A (f) A f ( ) s A s () A (f)a () ( ) A () A (f) A f ( ) = A s () A (f)a () ( ) A f = A (f) ( ) A (f)a () Consequence 5. Let E L nd A e such tht the conditions L L A A e stisfied. If 0 < < s = f f f L f is positive A(f) > 0 nd A(f) nd in ddition A is nomlised then ( ) A f A(f ) A (f) ( ) A (f) when < < while if 0 < the tems nd exchnge thei positions in the peceeding inequlities. As n ppliction fo time scles integls of Theoem 9 we otin: Consequence 6. (i) Let T nd 0 < <. If f C d (T R) is positive nd then ( f (x) x f(x) x ( ) ) ( f(x) x ( ) f (x) x ( ( f(x) x when < < while if 0 < the tems nd exchnge thei positions in the peceeding inequlities. (ii) In the cse of the Cuchy nl time-scles integls nd the Cuchy α-dimond time scle integls simily inequlities cn e stted s ove. s

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