On Inclusion Relation of Absolute Summability
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- Ἐμμανουήλ Αρβανίτης
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1 It. J. Cotemp. Math. Scieces, Vol. 5, 2010, o. 53, O Iclusio Relatio of Absolute Summability Aradhaa Dutt Jauhari A/66 Suresh Sharma Nagar Bareilly UP) Idia aditya jauhari@rediffmail.com Pragati Siha S.M. PG) College, Chadausi, U.P.) Preselty worig i: Dept. of Mathematics Sarottam Istitute of Techology ad Maagemet Greater Noida, U.P., Idia Abstract A theorem cocerig some ew absolute summability method is proed. May other results some ow ad uow are deried. Keywords: Summability, Absolute summability, Iclusio relatio 1 INTRODUCTION Let a be a ifiite series with partial sums s ad let q be a sequece with q 0 > 0, q 0, for >0 ad = q =0 we defie q = =0 A 1 q, = q, i = q i =0,i 1) 1.1) =0 where A 0 =1,A + 1) +2)... + ) = > 1,=1, 2, 3...). 1.2)!
2 2642 A. D. Jauhari ad P. Siha The sequece to sequece trasformatio [10] t = 1 q s 1.3) defies the sequece t ) of the N,q ) mea of the sequece s, geerated by the sequece of coefficiet q). Deoted by A d,g, the sequece space defied by ) A δ, γδ 1)+ = s ); a < ; a = s s 1 1.4) ad =0 =1 A f,d,g = s ); q =1 φ γδ 1)+ ) a < ; a = s s 1 1.5) Where φ is a sequece of possitie umber, 1, δ 0 ad γ, a real umber. Assume α β ad p be sequeces of positie umbers such that P = p =0 A lots of wor has already bee doe for absolute summability ad iclusio relatios. [4], [6], [7], [8], [9], etc. Recetly a theorem o iclusio was proed by SULAIMAN [11]. The purpose of this paper is to geeralize the aboe theorem ad to obtai some ew results. Our theorem is as follows :- THEOREM Let t deote the N,q,δ,γ) mea of the series a. If ad =+1 α β =1 α T = β 1+γδ 1/) Δt 1 p ) α γδ 1)+ p = O ) 1 1.6) P) P 1 P ) ) γδ 1)+ ) ) p ε T < 1.7) q P ) γδ 1)+ ε T < ad 1.8) β =1 α =1 β ) γδ 1)+ ) 1 q Δε T < 1.9)
3 O iclusio relatio of absolute summability 2643 the the series a ε is summable N,p,α,δ,γ, 1,δ 0,γ, a real umber. Proof of Theorem Let τ be the N,p ) mea of the series a ε, the p Δτ 1 = PP 1 = p PP 1 = p P P 1 P 1a ε P 1 1 1) 1 a ε ) ) 1a r Δ P ε + = p ) 1 PP β 1 γδ 1/) T 1 q [ q P 1 ε + p ) 1 1) ε + P ) ) + p ε β 1 γδ 1/) T q = p PP 1 1 q P β 1 γδ 1/) ) ) + p q P P T Δε + q ε β 1/ 1 δ T = T,1 + T,2 + T,3 + T,4, say r=1 ) ) 1 1a r 1 P 1ε ] ) 1 1 Δε ) β 1 γδ 1/) p T ε + β 1 γδ 1/) P T ε To proe the theorem, by Miowsi s iequality, it is sufficiet to show that =1 T,r <,r =1, 2, 3, 4
4 2644 A. D. Jauhari ad P. Siha Applyig Hölder s iequality, we hae- T,1 = α γδ 1)+ p PP 1 =2 1 q ) ) 1 1 ) ) 1 P p =2 =2 O1) O1) p P P P p 1 m m 1 q 1 q ) ) P p p q Δε β 1 γδ 1/) ) ) γδ 1)+ α Δε T β < by 1.9) =2 =2 O1) O1) T,2 = p =2 ) 1 P P P p 1 m ) p q m ) α q p α γδ 1)+ PP 1 ) p ) 1 ε β 1 γδ 1/) β < by 1.7) =2 T,3 = p P = P p 1 O1) O1) m ) P p p m α β q T =+1 p ) γδ 1)+ ) p P ε T =2 ) 1 P 1 ) 1 P ε β 1 γδ 1/) ) γδ 1)+ ε T < by 1.8) m α δ+ 1 T = =2 O1),4 =1 m ) α q β < by 1.7) ) γδ 1)+ p P T ) 1 P p ) p Δε β 1 γδ 1/) T Δε β 1 γδ 1/) T =+1 q ε β 1 γδ 1/) T p 1 P PP 1 p p T ) p ) 1 P P 1 p ε β 1 γδ 1/) T ) p ) 1 P P 1 ) p ε β 1 γδ 1/) T ) p ε β 1 γδ 1/) T =+1 ) ) p q P ) ε T ) p ) 1 P P 1 ε β 1 γδ 1/) T
5 O iclusio relatio of absolute summability 2645 COROLLARIES 1. For =0,δ =0 ad γ =1,Our theorem reduces to the theorem of SU- LAIMAN [11]. 2. For = 0 ad γ = 1 it gies a ew results, for N,q,δ), which implies N,q,α,δ. 3. For =0,δ = 0 ad γ = 1 ad ε = 1, it gies the results of BOR [1] ad [2]. 4. If α =,β = P p, =0,δ = 0 ad γ = 1 the we hae result of BOR & THORPE [5]. Acowlegemet Author are thaful to Dr. R.K. Sriastaa Associate Professor) Dept. of Mathematics, Agara College Agra. Refereces [1] BOR H. A ote o some absolute summability methods, J. Nigerio Math. Soc ) [2] BOR H. O two summability methods, Math. Proc. Cambridge Philos Soc ), [3] BOR H. A ote o two summability methods. Proc. Amer. Math. Soc., ), [4] BOR H. & ÖZARSLAN, H.S. O the uasi Mootoe ad almost icreasig sequeces, Joural of Math. Iequalities Vol 1) ), [5] BOR H. & THORPE, B. O some absolute summability methods, Aalysis ), [6] DAS G. Tauberia theorem for absolute Nörlud summability, Proc.Lodo Math. Soc. 3) ), [7] F. MORICZ & RHOADES, B. Necessary ad sufficiet coditio Tauberia coditios for certai weighted mea methods of summability II), ActaMath. Hugar )2004),
6 2646 A. D. Jauhari ad P. Siha [8] Seyha, A.H. A ote o, p α summability factors, Soochow Joural of Math. Vol. 27 1) 2001), [9] SINGH, N. & Sharma, N. O product of double sequeces, Bull. Cal. Math. Soc., 93 2) 2001) [10] SHARMA D. K. Ph.D. Thesis, M.J.P. Rohilhad Ui. 2002). [11] SULAIMAN W.T. O a ew absolute summability method, Iterat. J.Math. & Math. Sci. Vol. 21 No ) Receied: July, 2010
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