By Y. Okuyama

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Okuyama in Chapter Theorem dition [59] proved the following theorem, which will be 6. 7. Let ~(t) be a non-decreasing function. 2) {~(n)~(n)/n} function. 1). 5. 5 in form. 8. ~(t) , t > 0, t~'(t)/~2(t) is equivalent Let is a {pn } be non-negative positive is non-increasing, is non-increasing, and non-increasing. 9) ' 56 k= n l(k)lJ(k) k Pk _ 0 (l(n____~) ) , n = l , n 2 .... 11) C(> ~) then the series k(n)~(n)An+l(t) n=l is summable IN,Pnl , at t =x. If we put in our theorem (~ ~ 0), then tk'(t)/k2(t) t2k'(t)/k2(t) =at/(log k(t) =~/(log = (log t) e and u(k) t) ~+I = i/(log k) ~ is non-increasing t) ~+I is non-decreasing.

N= I n is a generalization of the theorems of Singh [75] and [57]. sequence [89]. 5. 5) in m, n. Let {~(n)} be a positive series 0<~ i ~(n)/n n=l converges. 6) then the series ~ n~(n)An(t) n=l is summable IN,Pnl , at t =x. 5 is an extension the case B = 0. 6. 5. theorem. 8) and 2~/t)Idf(t) I < ~ J0 are mutually Also, proved exclusive. Okuyama in Chapter Theorem dition [59] proved the following theorem, which will be 6. 7. Let ~(t) be a non-decreasing function. 2) {~(n)~(n)/n} function. 1). 5. 5 in form.

K=n k Pk n Therefore, by T h e o r e m 5 . 6. 5 following holds. corollary. If I n(log log C/t)Id¢(t) I < ~ , 0 then the series A~t)/log(n+2) is summable IN,log log(n+2)/(n+2) log(n+2)I, n=0 at t = x. 4. Other Conditions. 4. 1) are known. [29]. Let {pn } be a positive non-increasing sequence and ~ > 0. 3) then the series This theorem Varshney [ A (t) is summable IN,Pnl, at t =x. n= I n is a generalization of the theorems of Singh [75] and [57]. sequence [89]. 5. 5) in m, n. Let {~(n)} be a positive series 0<~ i ~(n)/n n=l converges.

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