On a more accurate multidimensional Mulholland-type inequality
© Chen and Yang; licensee Springer 2014
Received: 17 March 2014
Accepted: 29 July 2014
Published: 22 August 2014
In this paper, by using the way of weight coefficients and technique of real analysis, a more accurate multidimensional discrete Mulholland-type inequality with the best possible constant factor is given, which is an extension of the Mulholland inequality. The equivalent form, the operator expression with the norm as well as a few particular cases are also considered.
KeywordsMulholland-type inequality weight coefficient equivalent form operator norm
where the constant factor is still the best possible.
In this paper, by using the way of weight coefficients and technique of real analysis, a more accurate multidimensional discrete Mulholland-type inequality with the best possible constant factor is given, which is an extension of (3). The equivalent form, the operator expression with the norm as well as a few particular cases are also considered.
2 Some lemmas
Then we have (7). □
Hence we prove that (12) is valid for . Therefore, we have (11). □
Then by (13), we obtain (14). □
where and .
- (i)we have(17)
- (ii)for , , setting , , we have(20)
Hence, we have (17). In the same way, we have (18).
Hence, we have (20) and (21). □
3 Main results and operator expressions
we have the following.
is the best possible ( is indicated by (15)).
Then by (17) and (18), we have (23).
and then . Hence, is the best possible constant factor of (23). □
which is equivalent to (23).
namely, , and then (27) follows.
Then by (27), we have (23). Hence (27) and (23) are equivalent.
By the equivalency, the constant factor in (27) is the best possible. Otherwise, we would reach a contradiction by (28) that the constant factor in (23) is not the best possible. □
With the assumptions of Theorem 2, in view of , we have the following definition.
Since the constant factor in (32) is the best possible, we have:
- (ii)Putting , , () and (), in (32), we have the following new inequality:(37)
In particular, for , , in (37), we can deduce (4). Hence, (23) is an extension of (4).
This work is supported by the National Natural Science Foundation of China (No. 61370186), and 2013 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University (No. 2013KJCX0140).
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