By Y. Okuyama

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1. [85]. Let w(x) be a positive and non-decreasing function of x over the interval [N,~]. 1) is not greater than A ~ n=l 1 ~+1 { ~[ (n-j+l)2e-2 n . 2 j=l { j=l [ ~ "'" + 1 na+l " [nl/2]+l "'" = S + T, say. 3) we get S <= A ][ 1 ~ n [n I/2 ~-lnl/2 { n=l n [ j=l ] X2 a~-i D lajl2 9j}1/2 [n I/2 ] < A [ n -3/2 { [ n=l j =i I aj 12 ~j11/2 < ~. Similarly we have 1 ! (n_j+l) i n a+l {j En I/2] 2 ~ - 2 j 2 )~2 f~- 1 J n < A [ 1.. 1. 3 and = I and I. 3~3. Approximation and Absolute Summability. Applying we shall show some generalization of known theorems.

However, not appeal to the result of Das and Srivastava we need (cf. [34], Theorem B) for proofs of their corollaries. 1. [52]. 2. < ~ , is summable If 0 < ~ < I, B >__O, and I n(log C/t)BId¢(t)l 0 then the series Ic,BI, at t = x, where 0 ~ m < B < i. ~ (log n)BAn(t) n=l < ~ , is summable IC,~], at t =x. 53 This corollary coincides to Bosanquet [8] for 8 = 0, and Mohanty [55] for 8 = i• respectively. Corollary 5 3. If i > a > O• B > 0 J~(log 0 C/t)BId~(t)I ~ + 8< i• and < ~ , then the series An(t) n=O {log(n+2) } I-8 is summable IN,I/(n+2){log(n+2)}~l, at t = x .

Of these theorems. 3. Let {pn] be non-negative {~n } is a positive non-decreasing bounded function sequence such that [ p -~kXkk k = n k and non-increasing. and l(t), {~nXn/(n+l)} = 0 ( Suppose t > 0, is a positive is non-increasing, ), n = l , 2 .... 1) n and I T X(C/t)Id¢(t)l 0 < ~ , for a constant C(>2w). 2) Then the series n=O is summable IN,Pnl , at t =x. 1. 1. [17]. 2. 3. then for any x, n-i k+l [ APkX k=m of 49 where m and n are integers such that n > m > 0. This lemma is easily obtained.