Acta Scientiarum Universitatis Pekiniensis (Naturalum)

50 =| Ge ese ZS ee ec 1958 4¢

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ON THE MEAN-VALUE THEOREMS OF Z,,,(s) Min Szu hoa and Yin Wen lin (Department of Mathematics and Mechanics)

Apsrracr

One of the authors stated and proved some mean-value theorems for Z,,,(s).7 -

Since he quoted a “theorem” (theorem 74 of) which is erroneous, both the proofs

and the theorems have to be modified. The main cbject of the present paper is

to prove the following mean value theorems:

; ; 1 Theorem 1. If n is an even integer n>2, v= : and 0<a<kv—v, then y

co

I=—2— P| Z, , (w+ it) Pedi =

Vi Q0

0 = O (8-2-1) Ce) ane O (o“ a waa) ; (€>0) If further we have n>2 and a<kv—2kv?—v+2v", then 1 L= (20) 7¢,8?0V@ "94 (1 +0(1))

where ¢c, is @ constant, depending only on &, n and a.

; i Alin 5 Theorem 2. If nm is an even integer, n>2, v=— and 0<a<hv then n

lg JV Zan att) Pa =O (ENE #)-042) LO (PMO) (E>) 0

If further we have n>2 and a<kv—2kv?—v+2v?, then J\ Zan lait) [dt ~ og TAH ar—t-aynzaes ,

where cs; is @ constant depending only on &, n and a. Moreover, let w(o7) =p,(o7) and v(a7) =v,(c) be respectively the greatest lower bound of € and that of €' for which

Zy4 (7 +0) =O(\t °)

and il T ps | 2ne(o+ét) Pai=O(\4/"). 0

Then, the following conjecture may be of interest: 1

(o) =ty (7) =max {9, kv-ao——, (n—1) (kv=y —o) -o+5}.

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