A new reverse Mulholland’s inequality with one partial sum in the kernel

By means of the techniques of real analysis, applying some basic inequalities and formulas, a new reverse Mulholland’s inequality with one partial sum in the kernel is given. We obtain the equivalent conditions of the parameters related to the best value in the new inequality. As applications, we reduce to the equivalent forms and a few inequalities for particular parameters

In this article, following the methods of [16,17], by means of the techniques of analysis, several basic inequalities and formulas, a new reverse Mulholland's inequality with one partial sum in the kernel is given.The equivalent conditions of the parameters related to the best value in the new inequality are obtained.We also deduce the equivalent forms and a few equivalent inequalities for particular parameters.
Lemma 4 For t > 0, the following inequality is valid: A m e -t ln me -t ln(m+1) A m e -t ln me -t ln(m+1) .
This proves lemma.
This proves the lemma.

Theorem 1
The following reverse Mulholland's inequality with A m in the kernel is valid: In particular, for λ 1 + λ 2 = λ, we have and the following reverse inequality: Proof In view of the following expression related to the Gamma function: t λ e -(ln m+ln n)t dt, by ( 14), it follows that Then by (15), in view of (λ + 1) = λ (λ), we have (17).For λ 1 + λ 2 = λ in (17), we have (18).This proves the theorem.
is the best possible.
Proof We now show that 1 λ B(λ 1 , λ 2 ) in ( 18) is the best value under the assumptions of this theorem.
We obtain 1) 1) In view of (11) (for Then we have .
Setting ε → 0 + , in view of the continuity of the Beta function, we obtain is the best value in (18).This proves the theorem.

Equivalent forms and some particular inequalities
Theorem 4 The following reverse inequality equivalent to (17) is valid: Particularly, for λ 1 + λ 2 = λ, the following reverse inequality equivalent to (18) is valid: Proof Assuming that ( 23) is valid, by the reverse Hölder's inequality, we have In view of (23), we have (17).Assuming that ( 17) is valid, we set Then we find If J = ∞, then ( 23) is valid; if J = 0, then it is impossible that makes (23) valid, namely, J > 0. Assuming that 0 < J < ∞, by (17), it follows that .
is the best possible.On the other hand, if the same constant in (23) is the best possible, then for λ - Proof We show that the constant 1 λ B(λ 1 , λ 2 ) in ( 24) is the best possible.Otherwise, by (25) (for λ 1 + λ 2 = λ), we would reach a contradiction that the same constant in (18) is not the best possible.
On the other hand, if the constant in ( 23) is the best possible, then the same constant in (17) is also the best possible.Otherwise, by (26) (for λ 1 + λ 2 = λ), we would reach a contradiction that the same constant in (24) is not the best possible.
This proves the theorem.

Conclusions
In this article, by means of the techniques of analysis, applying the basic inequalities and formulas, a new reverse Mulholland's inequality with one partial sum in the kernel is given in Theorem 1.The equivalent conditions of the best value related to parameters are obtained in Theorems 2 and 3.As applications, we deduce the equivalent forms in Theorems 4 and 5, and some new inequalities for particular parameters in Remark 1.

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Proof Since A m e -t ln m = o(1) (m → ∞), by Abel's summation by parts formula, it follows that ∞ m=2 e -t ln m a m = lim m→∞ A m e -t ln m + ∞ m=2