• Korean Journal of Mathematics
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• v.23 no.4
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• pp.655-664
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• 2015

In this paper, we prove the discrete Hardy inequality through the continuous case for decreasing functions using elementary properties of calculus. Also, we prove the Carleman's inequality through limiting the discrete Hardy inequality with applications of Taylor series.

• Kyungpook Mathematical Journal
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• v.46 no.3
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• pp.425-431
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• 2006

In this paper, by introducing three parameters A, B and ${\lambda}$, and estimating the weight coefficient, we give a new extension of Hardy-Hilbert's inequality with a best constant factor, involving the Beta function. As applications, we consider its equivalent inequality.

• Kyungpook Mathematical Journal
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• v.47 no.3
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• pp.411-423
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• 2007

This paper deals with a reverse Hardy-Hilbert's inequality with a best constant factor by introducing two parameters ${\lambda}$ and ${\alpha}$. We also consider the equivalent form and the analogue integral inequalities. Some particular results are given.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.2
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• pp.231-237
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• 2018

Hardy's inequality is refined non-trivially as the form $${\int_{0}^{{\infty}}}\{{\frac{1}{x}}{\int_{0}^{x}}f(t)dt\}^pdx{\leq}Q_f{\times}({\frac{p}{p-1}})^p{\int_{0}^{x}}f^p(x)dx$$ for some $Q_f:0{\leq}Q_f{\leq}1$. $P{\acute{o}}lya$-Knopp's inequality is also refined by the similar form.

• Kyungpook Mathematical Journal
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• v.50 no.1
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• pp.131-152
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• 2010

In this paper, by introducing some parameters we establish an extension of Hardy-Hilbert's integral inequality and the corresponding inequality for series. As an application, the reverses, some particular results and their equivalent forms are considered.

• Bulletin of the Korean Mathematical Society
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• v.38 no.4
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• pp.701-708
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• 2001

In this article, using the properties of power mean, some new generalizations of the strengthened Hardy's inequalities are given.

• Bulletin of the Korean Mathematical Society
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• v.39 no.1
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• pp.81-87
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• 2002

The weighted Hardy＇s inequality is known as (equation omitted) where -$\infty$$\leq$a$\leq$b$\leq$$\infty$ and 1 < p < $\infty$. The purpose of this article is to provide a useful formula to express the weight r(x) in terms of s(x) or vice versa employing a Bernoulli equation having the other weight as coefficients.

• Kyungpook Mathematical Journal
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• v.49 no.3
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• pp.493-506
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• 2009

By using the improved Euler-Maclaurin summation formula and estimating the weight coefficients in this paper, a new dual Hardy-Hilbert's inequality and its reverse form are obtained, which are all with two pairs of conjugate exponents (p, q); (r, s) and a independent parameter ${\lambda}$. In addition, some equivalent forms of the inequalities are considered. We also prove that the constant factors in the new inequalities are all the best possible. As a particular case of our results, we obtain the reverse form of a famous Hardy-Hilbert's inequality.

• Kyungpook Mathematical Journal
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• v.52 no.4
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• pp.397-404
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• 2012

In this article, the reverse q-H$\ddot{o}$lder type inequality and two dimensional q-H$\ddot{o}$lder's inequality are established. We also obtain some q-integral inequalities by using q-H$\ddot{o}$lder's inequality which give q-Hardy's inequalities as spacial cases.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.3
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• pp.321-324
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• 2018

We present a new and simple proof of improved Carleson's inequality.