On a more accurate Hardy-Mulholland-type inequality

By using the way of weight coefficients, the technique of real analysis, and Hermite-Hadamard’s inequality, a more accurate Hardy-Mulholland-type inequality with multi-parameters and a best possible constant factor is given. The equivalent forms, the reverses, the operator expressions and some particular cases are considered.


Introduction
Assuming that p > ,  where the constant factor π sin(π /p) is still the best possible. Also we have the following Mulholland's inequality similar to () with the same best possible constant factor π sin(π /p) (cf. (  ) For μ i = υ j =  (i, j ∈ N), inequality () reduces to ().
In , Yang [] gave an extension of () as follows: If  < λ  , λ  ≤ , λ  +λ  = λ, {μ m } ∞ m= and {υ n } ∞ n= are positive and decreasing, with U ∞ = V ∞ = ∞, then we have the following inequality with the best possible constant factor π/ sin( πλ  λ ): In this paper, by using the way of weight coefficients, the technique of real analysis, and Hermite-Hadamard's inequality, a new Hardy-Mulholland-type inequality with a best possible constant factor is given as follows: If μ  = υ  = , {μ m } ∞ m= and {υ n } ∞ n= are positive and decreasing, with U ∞ = V ∞ = ∞, we have the following inequality: which is an extension of (). Moreover, the more accurate inequality of () and its extension with multi-parameters and the best possible constant factors are obtained. The equivalent forms, the reverses, the operator expressions and some particular cases are considered.

Some lemmas and an example
In the following, we agree that p = , ,  , and f (x) is strictly increasing in the intervals (n -  , a), (a, n) and (n, n +   ), respectively, satisfying then we have the following Hermite-Hadamard's inequality (cf. []).
Proof In view of f (n -) ≤ f (n + ) = lim x→n + f (x) is finite, we set the linear function g(x) as follows: Since f (x) is strictly increasing in [n -  , a) and (a, n), then for x ∈ [n -  , a), In the same way, since f (x) is strictly increasing in (n, n +   ), then for x ∈ (n, n +   ), f (x) > f (n + ) ≥ f (n -). Hence, Therefore, we have f (x)g(x) > , x ∈ (n -  , n +   )\{n}. Then we find namely, () follows. The lemma is proved.
Note With the assumptions of Lemma , if (i) a ∈ (n, n +   ), f (x) is strictly increasing in the intervals (n -  , n), (n, a) and (a, n +   ), respectively, or (ii) a = n, f (x) is strictly increasing in the intervals (n -  , n) and (n, n +   ), respectively, then in the same way, we still can obtain ().
(i) For γ = , we obtain (ii) for - < γ < ,  < +γ -γ < , by the Lebesgue term by term integration theorem (cf. []), we find For fixed m ∈ N\{}, we define the function f (y) as follows: Then f (y) is continuous in (n -  , n +   ) (n ∈ N\{}). There exists a unified number For y  = n, we obtain for y = n that for y  = n, we obtain for y = n that Since for y  = n, V (y  -) = υ n , V (y  + ) = υ n+ and for y  = n, V (y  -) = V (y  ), then We still can find that y=n , and f (y) (< ) is strictly increasing in (n -  , n) and (n, n +   ). Therefore, f (y) satisfies the conditions of Lemma  with Note. So does g(y) = f (y) V (y) ln -λ  βV (y) . Hence, by (), we have Definition  Define the following weight coefficients: () Lemma  If {μ m } ∞ m= and {υ n } ∞ n= are decreasing and U ∞ = V ∞ = ∞, then for m, n ∈ N\{}, we have the following inequalities: where K γ (λ  ) is determined by ().
For c > , we find Hence, we obtain (). In the same way, we obtain ().
Proof We find, for  < δ < min{λ  , λ  }, In the same way, we find and then we have ().

Main results
In the following, we also set () Theorem  (i) For p > , we have the following equivalent inequalities: (ii) for  < p <  (or p < ), we have the equivalent reverse of () and ().
[]) and (), we obtain the reverse of () (or ()), then we have the reverse of (), and then the reverse of () follows. By Hölder's inequality (cf. []), we have the reverse of (), and then by the reverse of (), the reverse of () follows.
On the other hand, assuming that the reverse of () is valid, we set b n as (). Then we find J p = b q q,˜ λ . If J = ∞, then the reverse of () is trivially valid; if J = , then by the reverse of (), () takes the form of equality (= ). Suppose that  < J < ∞. By the reverse of (), it follows that the reverses of () and () are valid, and then the reverse of () follows, which is equivalent to the reverse of ().
Theorem  If p > , {μ m } ∞ m= and {υ n } ∞ n= are decreasing, U ∞ = V ∞ = ∞, a p, λ ∈ R + and b q q, λ ∈ R + , then we have the following equivalent inequalities: where the constant factor K γ (λ  ) is the best possible.
Theorem  If p < , {μ m } ∞ m= and {υ n } ∞ n= are decreasing, U ∞ = V ∞ = ∞, a p, λ ∈ R + and b q q, λ ∈ R + , then we have the following equivalent inequalities with the best possible