• Journal of the Korean Mathematical Society
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• v.36 no.5
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• pp.855-890
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• 1999

Necessary and sufficient conditions on the weight functions u(.) and $\upsilon$(.) are derived in order that the fractional maximal operator $M\alpha,\;0\;\leq\;\alpha\;<\;1$, is bounded from the weighted amalgam space $\ell^s(L^p(\mathbb{R},\upsilon(x)dx)$ into $\ell^r(L^q(\mathbb{R},u(x)dx)$ whenever $1\leq s\leq r<\infty\;and\;1. The boundedness problem for the fractional intergral operator $I_{\alpha},0<\alpha\leq1$, is also studied.

• Korean Journal of Mathematics
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• v.14 no.2
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• pp.137-160
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• 2006

We describe the weight functions for which Hardy's inequality of nonincreasing functions is satisfied. Further we characterize the pairs of weight functions $(w,v)$ for which the Laplace transform $\mathcal{L}f(x)={\int}^{\infty}_0e^{-xy}f(y)dy$, with monotone function $f$, is bounded from the weighted Lebesgue space $L^p(w)$ to $L^q(v)$.

• Journal of the Chungcheong Mathematical Society
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• v.28 no.4
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• pp.583-590
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• 2015

This paper deals with linear impulsive fractional differential equations involving the Caputo derivative with non-integer order q. We provide exact solutions of linear impulsive fractional differential equations with constant coefficient by mean of the Mittag-Leffler functions. Then we apply the exact solutions to improve impulsive integral inequalities with singularity.

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.619-630
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• 2018

In this paper, we obtain a (p, q)-extension of the Whittaker function $M_{k,{\mu}}(z)$ together with its integral representations, by using the extended confluent hypergeometric function of the first kind ${\Phi}_{p,q}(b;c;z)$ [recently extended by J. Choi]. Also, we give some of its main properties, namely the summation formula, a transformation formula, a Mellin transform, a differential formula and inequalities. In addition, our extension on Whittaker function finds interesting connection with the Laguerre polynomials.

• Journal of applied mathematics & informatics
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• v.33 no.3_4
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• pp.275-292
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• 2015

This paper is mainly concerned with the existence of solution for nonlinear impulsive fractional dynamic equations on a special time scale.We introduce the new concept and propositions of fractional q-integral, q-derivative, and α-Lipschitz in the paper. By using a new fixed point theorem, we obtain some new existence results of solutions via some generalized singular Gronwall inequalities on time scales. Further, an interesting example is presented to illustrate the theory.

• Journal of the Korean Mathematical Society
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• v.45 no.1
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• pp.63-78
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• 2008

We generalize several integral inequalities for analytic functions on the open unit polydisc $U^n={\{}z{\in}C^n||zj|<1,\;j=1,...,n{\}}$. It is shown that if a holomorphic function on $U^n$ belongs to the mixed norm space $A_{\vec{\omega}}^{p,q}(U^n)$, where ${\omega}_j(\cdot)$,j=1,...,n, are admissible weights, then all weighted derivations of order $|k|$ (with positive orders of derivations) belong to a related mixed norm space. The converse of the result is proved when, p, q ${\in}\;[1,\;{\infty})$ and when the order is equal to one. The equivalence of these conditions is given for all p, q ${\in}\;(0,\;{\infty})$ if ${\omega}_j(z_j)=(1-|z_j|^2)^{{\alpha}j},\;{\alpha}_j>-1$, j=1,...,n (the classical weights.) The main results here improve our results in Z. Anal. Anwendungen 23 (3) (2004), no. 3, 577-587 and Z. Anal. Anwendungen 23 (2004), no. 4, 775-782.