SOME INTEGRAL INEQUALITIES FOR HARMONICALLY CONVEX STOCHASTIC PROCESSES ON THE CO-ORDINATES

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1 Avne Mth Moels & Applitions Vol3 No 8 pp63-75 SOME INTEGRAL INEQUALITIES FOR HARMONICALLY CONVE STOCHASTIC PROCESSES ON THE CO-ORDINATES Nurgül Okur * Imt Işn Yusuf Ust 3 3 Giresun University Deprtment of Sttistis Giresun Turkey Giresun University Deprtment of Mthemtis Giresun Turkey Astrt The purpose of this pper is to introue hrmonilly onvex stohsti proesses on the oorintes in orer to generlize the lssil onvex stohsti proesses n to otin new estimtions Hermite-Hmr type ineulities n estimtion for hrmonilly onvex stohsti proesses with this purpose in min re otine These results re prtiulrly interesting from optimiztion view point sine it provies roer setting to stuy the optimiztion n mthemtil progrmming prolems n to ompre the mximum n minimum vlues of stohsti proess Keywors: Hrmonilly onvex stohsti proess on o-orintes men-sure ifferentile mensure integrl Hermite-Hmr ineulity AMS Sujet Clssifition: Primry: 6D5; Seonry: 6D99 6A5 6N Corresponing Author: Nurgül Okur Giresun University Deprtment of Sttistis Fulty of Sienes n Letters Güre Cmpus 8 Giresun Turkey e-mil: nrgokur@gmilom Mnusript reeive: 98; Revise 8; Revise 58; Aepte 88 Introution hve It is well known tht for every rel onvex funtion f on the intervl we f f(x)x f f These re elerte Hermite-Hmr ineulities In proilisti wors they sy tht f E x Ef x Ef f C x where E enotes mthemtil expettion (respetively ) is rnom vrile hving the uniform istriution on the intervl (respetively on the set ) C x is the set of ll rel onvex funtions on n x stns for the so lle onvex orer of rnom vriles (De l Cl et l 6) There re mny stuies in reent yers on some types of onvexity for stohsti proesses n Hermite-Hmr ineulities for relte onvex stohsti proesses n it is of gret importne in optimiztion espeilly in optiml esigns n lso useful for numeril pproximtions when there exist proilisti untities in the literture Convex stohsti proesses were propose n some properties were given for lssil onvex stohsti proessesy Nikoem (98) Stohsti onvexity n its pplitions were efine y Shke et l (988) Jensen-onvex λ-onvex 63

2 ADVANCED MATH MODELS & APPLICATIONS V3 N 8 stohsti proesses were introue y Skowronski (99)The lssil Hermite- Hmr ineulity to onvex stohsti proesses ws extene y Kotrys () Convex stohsti proesses on the o-orintes were introue n Hermite- Hmr type ineulities for these proesses were otine y Set et l (5) Hrmonilly onvex stohsti proesses were efine n Hermite-Hmr type ineulities were otine y Okur et l (8) The uthors finings le to our motivtion to uil our work The min sujet of this pper is to pt some otine results onerning hrmonilly onvex funtions on the o-orintes y Noor et l (5) To hrmonilly onvex stohsti proesses on the o-orintes n to otin Hermite-Hmr type ineulities n estimtion for these proesses Thus the hrmonilly onvex stohsti proesses whih efine y Okur et l (8) is extenteon two-imensionl intervl in this stuy Preliminries Let us rell some importnt types of onvexity for stohsti proesses: Definition (Kotrys ) Let (Ω I P) e n ritrry proility spe A funtion : Ω R is lle rnom vrile if it is I-mesurle : I Ω R where I R is n intervl is lle stohsti proess if for every t I the funtion (t ) is rnom vrile (Kotrys ) Definition (Kotrys ) Let (Ω I P) e n ritrry proility spe n I R e n intervl The stohsti proess : I Ω R is lle lmost everywhere (i) onvex if for ll t s I n λ (ii) λ-onvex if λt ( λ)s λ t ( λ)(s ) λt ( λ)s λ t ( λ)(s ) for ll t s I n λ is fixe numer in (iii) Jensen-onvex if t s t s for ll t s I (Kotrys) Let us give some si efinitions n notions out ontinuity onepts n ifferentiility for stohsti proesses n men-sure integrl of stohsti proess Definition 3 (Kotrys ) Let (Ω I P) e n ritrry proility spe n I R e n intervl We sy tht stohsti proess : I Ω R is lle (i) ontinuous in proility on I if for ll t I if P lim t t t t where P lim enotes limit in proility 6

3 N OKUR et l: SOME INTEGRAL INEQUALITIES FOR HARMONICALLY CONVE (ii) men-sure ontinuous on I if for ll t I if lim E t t t t where E[ t ] enotes expettion vlue of the rnom vrile t (iii) inresing (eresing) if for ll t s I suh tht t< s if t s t s (iv) monotoni if it is inresing or eresing (v) men-sure ifferentile t point if t I if there is rnom vrile t : Ι Ω R suh tht t P lim t t t t t t We sy tht stohsti proess : Ι Ω R is ontinuous (ifferentile) if it is ontinuous (ifferentile) t every point of the intervl I (Kotrys ) Definition (Kotrys ) Let (Ω I P) e n ritrry proility spe n I R e n intervl n : Ι Ω R e stohsti proess with E t for ll t I Let I t < t t n e prtition of n Θ k [t k t k ] ritrry for k n A rnom vrile Y: Ω R is lle men-sure integrl of the proess t on if the following ientity hols: lim E n n k Then we n write lmost everywhere Θ k t k t k Y t t Y The men-sure integrl opertor is inresingon lmost everywhere tht is t Z t t t Z t t Now we rell the well-known Hermite-Hmr integrl ineulity for onvex stohsti proesses: Theorem (Kotrys ) If : I Ω R is Jensen-onvex n men sure ontinuous in the intervl Ι Ω then for ny I < we hve lmost everywhere t t Also Set et l (5) estlishe the following similr ineulity of Hmr s type for o-orinte onvex stohsti proesses on retngle from the plne R : 65

4 ADVANCED MATH MODELS & APPLICATIONS V3 N 8 Definition 5 (Set et l 5) Let us onsier two-imensionl intervl in R with < n < A stohsti proess : Ω R is si to e onvex on Ω if the following ineulity hols lmost everywhere λt λ t λs λ s λ t s λ t s for ll (t s ) (t s ) n λ If the ove ineulity is reverse then is si to e onve on (Set et l 5) Theorem (Set et l 5) Suppose tht : Ω R is o-orinte onvex on Then the following ineulities re true lmost everywhere (Set et l 5): t t t s ts s) s t t t s s s The ove ineulities re shrp Let us onsier the Hermite-Hmr integrl ineulity for hrmonilly onvex stohsti proesses: Definition 6 (Okur et l 8) Let I R e rel intervl A stohsti proess : I Ω R is si to e hrmonilly onvex stohsti proess lmost everywhere if ts λ s λ t λt ( λ)s for ll t s I n λ If the ove ineulity is reverse then is si to e hrmonilly onve lmost everywhere (Okur et l 8) The following result of the Hermite-Hmr type ineulities hols: Theorem 3 (Okur et l 8) Let I R e rel intervl n : I Ω R e hrmonilly onvex stohsti proess n I with < If L[ ] then the following ineulities hollmost everywhere 66

5 N OKUR et l: SOME INTEGRAL INEQUALITIES FOR HARMONICALLY CONVE t t t The ove ineulities re shrp (Okur et l 8) () 3 Min Results Motivte y Set et l (5) n s ontriution to Okur et l (8) we explore new onept of onvex stohsti proesses n introue prtiulrly hrmonilly onvex stohsti proesses on the o-orintes in present stuy By virtue of this new onept we present Hermite-Hmr ineulities n n interesting importnt estimtion for these stohsti proesses Throughout this setion let us onsier the two-imensionl intervl [ ] in with < n < from here Definition 7 A stohsti proess : Ω R is si to e hrmonilly onvex on if the following ineulity holslmost everywhere t s t s λ t λt λ s λt λ s s λ t s for ll (t s ) (t s ) n λ [] If the ove ineulity is reverse then is si to e hrmonilly onve on Definition 8 A stohsti proess : Ω R is si to e hrmonilly onvex on the o-orintes on if the prtil mppings s : Ω R s u u s n t : Ω R t v ((t v) ) efine for ll t n s re hrmonilly onvex lmost everywhere Now we give forml efinition of oorinte hrmonilly onvex stohsti proesses: Definition 9 A stohsti proess : Ω R is si to e hrmonilly onvex stohsti proess on lmost everywhere if t s t s φt φ s θt θ s φθ t s φ θ t s φ θ t s φ θ t s for ll (t s ) (t s ) n φ θ Lemm Every hrmonilly onvex stohsti proess : Ω R is hrmonilly onvex on the o-orinteslmost everywhere Proof Suppose tht : Ω R is hrmonilly onvex stohsti proess on Consier t : Ω R t v t v Then for ll λ s s [ ] the following ineulity hols lmost everywhere 67

6 ADVANCED MATH MODELS & APPLICATIONS V3 N 8 t ( λs λ s ) t λs λ s λt λ t λs λ s λ( t s ) λ ( t s ) λ t s λ t (s ) whih shows the hrmonilly onvexity of t The ft tht s : [ ] Ω R s u u s) is lso hrmonilly onvex on for ll s [ ] goes likewise n we shll omit the etils Theorem 3 Suppose tht : Ω R is hrmonilly onvex on the o-orintes on Then the following ineulities hol lmost everywhere: t t ( )( ) t s ( t s )ts s s The ove ineulities re shrp t t t t s s s s () Proof Sine : Ω R is hrmonilly onvex on the o-orintes it follows tht the mpping t : Ω R t s t s is hrmonilly onvex on [ ] for ll t Then lmost everywhere t Tht is t s t s s s t t (t s) s Integrting this ineulity on we hve ( t ) ( t ) 68

7 N OKUR et l: SOME INTEGRAL INEQUALITIES FOR HARMONICALLY CONVE t t t ( )( ) t t t t ( t s )ts By similr rgument pplie for the mpping s : [ ] Ω R s u u s) we get s s s ( )( ) s s s s ( t s )ts Summing the ineulities (3) n () we get the seon n the thir ineulity in () By the ineulity () we lso hve t s t s t whih give y ition the first ineulity in () Finlly y the ineulity () we lso hve t t t t t t s s s s s s whih give y ition the lst ineulity in () The ove ineulities re shrp Inee if : Ω R t s Thus s (3) () t s t s λt λ s λt λ s 69

8 ADVANCED MATH MODELS & APPLICATIONS V3 N 8 λ t s λ t s for ll for ll (t s ) (t s ) n λ Therefore is hrmonilly onvex on We lso hve ( )( ) ( t s )ts whih shows us the ineulities () re shrp Lemm Let : Ω R e prtil ifferentile mpping on with < n < If L( ) then the following eulity hols lmost everywhere ( ) ( φ)( θ) (A φ ) A φ ( )( ) ( t s )ts t t t t s s s s where A φ φ φ n θ θ Proof By integrtion y prts we hve lmost everywhere ( ) ( φ)( θ) (A φ ) A φ φθ φθ ( ) ( θ) ( φ) θ A φ 7

9 N OKUR et l: SOME INTEGRAL INEQUALITIES FOR HARMONICALLY CONVE θ A φ φ θ B B B (5) ( ) ( θ) θ θ ( θ) A φ A φ φθ θ φ A φ φθ θ A φ A φ φ A φ φθ whih ompletes the proof t t t t s s s s ( )( ) ( t s )ts 7

10 ADVANCED MATH MODELS & APPLICATIONS V3 N 8 Theorem Let : Ω R e prtil ifferentile mpping on If L n > is hrmonilly onvex stohsti proess on the oorintes on then the following eulity hols lmost everywhere: ( )( ) ( t s )ts t t t t s s s s ( )( ) p p C C C 3 C where A φ φ φ θ θ n C F ; ; F ; ; C F ; ; F ; 3; C 3 F ; 3; F ; ; β is the Euler Bet funtion efine y C F ; 3; F ; 3; β t s Γ t Γ s Γ t s ϕt ( ϕ) s ϕ for ll t s > n F is the hypergeometri funtion efine y F ; ; γ β for ll > > γ < ϕ ( ϕ) ( γϕ) ϕ 7

11 N OKUR et l: SOME INTEGRAL INEQUALITIES FOR HARMONICALLY CONVE Proof From Lemm we hve lmost everywhere ( )( ) ( t s )ts t t t t s s s s ( ) ( φ)( θ) (A φ ) A φ φθ By using the well-known Höler ineulity for oule integrls if : Ω R is hrmonilly onvex stohsti proess on the o-orintes on then the ineulities hol lmost everywhere: ( )( ) ( t s )ts t t t t s s s s ( ) (A φ ) / p p (A φ ) φθ ( φ)( θ) p φθ A φ φθ p φ( θ) (6) 73

12 ADVANCED MATH MODELS & APPLICATIONS V3 N 8 φ θ ( φ)( θ) where n esy lultion gives φθ AB (7) F ; ; F ; ; AB (8) F ; ; F ; 3; AB (9) F ; 3; F ; ; AB () F ; 3; F ; 3; Hene if we use (7)-() in (6) we otin the esire result This ompletes the proof Conlusion In this pper we efine n importnt extension of onvexity for stohsti proesses whih is lle hrmonilly onvex stohsti proesses on the o-orintes We lso otin Hermite-Hmr type ineulities for hrmonilly onvex stohsti proesses on the o-orintes In proilisti wors it n e interprete riefly s follows: H O T x E T x A O T HC where is stohsti proess hving the hrmonilly onvexity on the o-orintes; T R ; H O E n A O re respetively efine s hrmonilly men expettion vlue rithmetilly men of the proess ; x stns for the so lle onvex orer of stohsti proesses; HC is the set of ll hrmonilly onvex stohsti proesses on the o-orintes Therefore we otin estimtion for hrmonilly onvex stohsti proesses on the o-orintes This result n e interprete proilistilly s follows: 7

13 N OKUR et l: SOME INTEGRAL INEQUALITIES FOR HARMONICALLY CONVE A O T E T x K A O T K R These ineulities re neessry to ompre some vlues of stohsti proess with its expete vlue These onepts my e prtiulrly interesting from optimiztion view point sine it provies roer setting for stuying optimiztion n mthemtil progrmming prolems As speil ses one n otin severl new n orret versions of the previously known results for vrious lsses of these stohsti proesses Applying some type of ineulities for stohsti proesses is nother promising iretion for future reserh Referenes De l Cl J Crmo J (6) Multiimensionl Hermite-Hmr ineulities n the onvex orer J Mth Anl Appl Drgomir SS Pere CEM () Selete topis on Hermite-Hmr ineulity n pplitions Vitori University Melourne Gller RG (3) Stohsti proesses theory for pplitions Cmrige University Press New York Işn I () Hermite-Hmr ineulities for hrmonilly onvex funtions Het J Mth Stt 3(6) Josip E Peri JE Proshn F Thong YL (99) Convex Funtions Prtil Orerings n Sttistil Applitions Mthemtis in Siene n Engineering Series Aemi Press Kotrys D () Hermite-Hmr ineulity for onvex stohsti proesses Aeut Mth Nikoem K (98) On onvex stohsti proesses Aeut Mth 8-97 Noor M Noor Khli Awn MU (5) Integrl ineulities for oorinte hrmonilly onvex funtions Complex Vriles n Ellipti Eutions 6(6) Okur N Işn I Yüksek Dizr E (8) Hermite-Hmr ineulities for hrmonilly onvex stohsti proesses Interntionl Journl of Eonomi n Aministrtive Stuies (8 EYI Speil Issue) 8-9 Set E Srıky MZ Tomr M (5) Hermite-Hmr ineulities for o-orintes onvex stohsti proesses Mthemti Aetern 5() Shke M Shnthikumr JG (988) Stohsti onvexity n its pplitions Avnes in Applie Proility 7-6 Skowronski A (99) On some properties of J-onvex stohsti proesses Aeut Mth