OSR Joual of Matheatics (OSR-JM) e-ssn: 2278-5728, p-ssn:239-765x. Volue 0, ssue Ve. V. (Feb. 204), PP 56-62 www.iosjouals.og A New Result O A,p,δ -Suabilty Ripeda Kua &Aditya Kua Raghuashi Depatet of Matheatics FTM Uiesity Moadabad, U.P., dia, 24400 Abstact: this pape we hae established a ew theoe o Ap,, ad iteestig esults ad peious ow esults as a coollay. Keywods:, N p -suability, A -suability, A, -suability, Ap,, ifiite seies. Matheatical classificatio :40D25, 40E05, 40F05 & 40C0.. NTRODUCTON: -suability which gies soe ew -suability ad Let a be a gie ifiite seies with the sequece of patial su ( s ) ad let A ( a ) be a oal atix of o zeo diagoal eties. The A defies the sequece to sequece tasfoatio appig the sequeces s ( s ) to As ( A( s)), A (s) A s whee The seies whee a is said to suable A, if (RHOADES ad SAVAS [3]) (.) A ( s) (.2) A ( s) A ( s) A ( s) ad it is said to be suable A,, 0 ad 0 if Let ( ) p be a sequece of positie ubes such that A (.3) P p as (.4) 0 whee Pi pi 0, i ad a is said to be suable A, p, if (ÖZARSLAN, [2]) P p Ad is said to be suable A, p,,. A ( s) if P A ( s) p (.6) p f P, 0, Ap,, -suability is the sae as A -suability also if we tae a, the Ap, -suability is the sae as N, p -suability (BOR []). A sequece ( b ) of positie ubes is said to be -quasi ootoe, if b 0 ultiately ad b whee ( ) is a sequece of positie ubes (SAVAS [4]). ad a sequece ( d ) is said to be alost iceasig if thee exist a positie iceasig sequece ( c ) ad two positie costats A ad B such that Ac d Bc fo each. (.5) p www.iosjouals.og 56 Page
A New Result O A,p,δ -Suabilty. KNOWN RESULT: Coceig with absolute atix suability facto SAVAS [5] has poed the followig theoe. Theoe 2. Let A be a lowe tiagula o Noal atix with o-egatie eties satisfyig a,0 (2.) a, fo a (2.2) a O() (2.3) whee A associates with two lowe tiagula atices A & Â defied. ad a O( a ) (2.4) a, O( ) (2.5) a a,, 0,,2 ad a a a,, 2,3, f ( X ) is a alost iceasig sequece such that, X X O ad (2.6) 0 as (2.7) Suppose that thee exist a sequece of ubes ( A ) such that it is -quasi X, AX is coeget ad t O X ( ) whee t a, the the seies a is suable A,, ad 0.. MAN RESULT: The goal of this pape is to geealize the theoe (2.) fo Ap,, -suability. Theoe 3. follows ootoe with f A ( a ) is ay oal atix associated with two lowe sub-atices A ( a ) ad A ( a ) a a,, 0,,2 ad, â a a. whee 0,0 0,0 0,0 f the coditios (2.8) (2.9) as i (3.) i a a a (3.2) a,0 (3.3) www.iosjouals.og 57 Page
a, a, fo ad let ( p ) be the sequece of positie ubes such that, P O( p ) as A New Result O A,p,δ -Suabilty (3.4) p a O P (3.5) P P a O a p p P P a, O p p P f { X } is a alost iceasig sequece such that X O( X ) p Suppose that thee exist a sequece of ubes ( A ) such that it is -quasi AX is coeget ad A fo all, if whee t a ae satisfied the the seies p P (3.6) (3.7) ad 0 as. X, ootoe with (3.8) P t O( X ) p (3.9) a issuable A, p,,, 0. V. LEMMA: We eed the followig leas fo the poof of theoe (3.). Lea 4.. Ude the coditio of theoe, we hae ((SAVAS [4]) X O() Lea 4.2 (SAVAS [5]) X is a alost iceasig sequece such that Let { } X X O f ( A ) is -quasi ootoe with X, A X is coeget, the X A X A ad O() (4.) Let { y } be the th te of the A -tasfo of V. PROOF OF THEOREM: iai the, i0 www.iosjouals.og 58 Page
A New Result O A,p,δ -Suabilty ad 0 Y a s i i i0 a a i i0 i0 0 i a a 0 a, i i a a (5.) y y y ( a a ) a, 0 a a (5.2) we ay wite a y a a a a a a a a 2 at a t a, ( ) t a, t ( ) T T T T (5.3) (say),,2,3,4 To coplete the poof, it is sufficiet, by Miowsi's iequality, to show that Usig Hölde's iequality ad (5.3), we get P T, p P T,, fo,2,3,4 p (5.4) P a t p P, O() a t p P a a p O() t P P a a p p O() t www.iosjouals.og 59 Page
P P a a t p p O() P P t a a, p p O() P O() a t p P P O() a t a t p p P P t t p p O() ( O() A X O() X A New Result O A,p,δ -Suabilty O(). (5.5) Agai, usig the hypothesis of the theoe (3.) ad Lea (4.), usig Hölde's iequality P 2 T,2 2 p P a ( ) t p 2, P a, t 2 p P, 2 p O() a t P,, 2 p O() a t a fo (Rhoades ad Saas[3]). Hece a Ma, P P 2, 2 p p O() a a t P P t a a, p p O() P t a, p O() P O() t p www.iosjouals.og 60 Page
A New Result O A,p,δ -Suabilty P P p t p p P O() P P P t t p p p O() O(). P O() X O() X p P O() X O() A X O() A X p O() Next usig the hypothesis of the theoe (3.) ad Hölde's iequality P 3 T,3 2 p Fially P p 2 a t, P a, t 2 p P t a, a, 2 p O() P P a t a, 2 p p O() P P a t a, p p O() P P t a a, p p O() P O() t a, p O() P t p P p O() ( t O() ( ) X O() X O() www.iosjouals.og 6 Page
A New Result O A,p,δ -Suabilty P 4 T,4 p P ( ) a t p P O() a t p P P O() a a t p p P O() a t p O(), as i the poof of. This copletes the poof of theoe. V. COROLLARY: This theoe hae the followig esults as a coollay. Coollay 6. P Taig the theoe (3.) educes to theoe (2.). p Coollay 6.2 P Taig, ad 0 the theoe (3.) is A -suable. p Coollay 6.3 p Taig a, ad 0 the theoe (3.) is N, p -suable. P REFERENCES: [] BOR, H; O local popety of N, p, -suability of factoed Fouie seies, J. Math. Aal. Appl. 79 (993). [2] OZARSLAN, H.S.; O absolute atix suability ethods, Matheatical Cou. 2, (2007). [3] RHOADES, B,E. ad SAVAS, E.; O A -suability factos, Alco, Math. Hug, (2006). [4] SAVAS, E.; A suability facto theoe fo absolute suability iolig -quasi -ootoe ad alost iceasig sequece, Math. Coput. Model, 48, (2008). [5] SAVAS, E.; A ote o absolute suability facto theoe ad alost iceasig sequeces, acadeic, Joual, Vol. 8 (203). www.iosjouals.og 62 Page