• The Pure and Applied Mathematics
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• v.15 no.4
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• pp.393-400
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• 2008

In this paper, we investigate the following additive functional inequality (0.1) ||f(x)+f(y)+f(z)+f(w)||${\leq}$||f(x+y)+f(z+w)|| in normed modules over a $C^*$-algebra. This is applied to understand homomor-phisms in $C^*$-algebra. Moreover, we prove the generalized Hyers-Ulam stability of the functional inequality (0.2) ||f(x)+f(y)+f(z)f(w)||${\leq}$||f(x+y+z+w)||+${\theta}||x||^p||y||^p||z||^p||w||^p$ in real Banach spaces, where ${\theta}$, p are positive real numbers with $4p{\neq}1$.

• Journal of Korean Physical Therapy Science
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• v.13 no.1
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• pp.21-27
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• 2006

The purpose of this study was to identify the above mentioned therapy on the reduction of functional leg length inequality, by the manipulation(Thonpson technique). In 8 patients who have been chronically ill with low back pain and functional leg length inequality, for past 12 weeks, we analyzed and observed the progress of symptom and sign on pelvis(femur head line level, ilium length, ilium rotation), using by X-ray. The results after 12 week treatment, compared with before treatment, were as follows : 1. The improved in femur head line in the manipulation after 12 week treatment was very significant(p<.01) 2. The improved in ilium length in the manipulation after 12 week treatment was very significant(p<.01) 3. The improved in ilium rotation in the manipulation after 12 week treatment was significant(p<.05).

• East Asian mathematical journal
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• v.23 no.2
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• pp.229-246
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• 2007

Some inequalities related to the $H{\ddot{o}}lder$ integral inequality and applications for bounding the ${\check{C}}eby{\check{s}}ev$ functional are given.

• Journal of Korean Medicine Rehabilitation
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• v.21 no.3
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• pp.13-20
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• 2011

Objectives : We studied the clinical utility of meridian electromyography for the assessment of erector spinae muscle in functional leg length inequality. Methods : We compared electrical activity between A group with a functional leg length inequality(n=17) and B group(n=23) in dynamic flexion-reextension state during five minutes. We anayzed amplitudes and areas of electrical activity and asymmetry index(AI). Results : 1. The short leg sides were significantly higher electrical activity than the long leg sides in the experimental group and control group(p<0.05). 2. The AI of A group significantly higher than B group(p<0.05). Conclusions : According to above results, there are correlations between erector spinae muscle and functional leg length inequality.

• Journal of the Korean Society of Physical Medicine
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• v.2 no.2
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• pp.183-194
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• 2007

Purpose : The purpose of this study was to evaluate influence of therapeutic exercise on functional leg length inequality(LLI). Methods : The subjects were consisted of thirty patients who had Leg length inequality(LLI) of more than 10mm(l6 females. 14 males) from 20 to 35 years of age(mean aged 24.23). All subjects randomly assigned to contract-relax exercise of proprioceptive neuromuscular facilitation(PNF) group(n=15), control group(n=15). Contract-relax exercise group received contract-relax exercise about 10 minutes with therapeutic massage about 15 minutes for 3 times per week during 4 weeks period. Control group not received intervention during 4 weeks period. The tape measure method(TMM) was used to measure functional Leg length inequality. Biodex System 3 Pro was used to measure strength of Knee extension & flexion. All measurements of each subjects were measured at pre-test, 2weeks post-test and 4weeks post-test. Results : 1. The LLI of contract-relax exercise group was significantly reduced according to within treatment period(p<.05), most significantly reduced between pre-test and post-test(p<.05). Contract-relax exercise group significantly more reduced than control group(p<.05). 2. The knee extension strength of contract-relax exercise group was significantly increased according to within treatment period(p<.05), most significantly increased between pre-test and post-test(p<.05). Contract-relax exercise group significantly more increased than control group(p<.05). 3. The knee flexion strength of contract-relax exercise group was significantly increased according to within treatment period(p<.05), most significantly increased between pre-test and post-test(p<.05). Contract-relax exercise group significantly more increased than control group(p<.05). Conclusion : Contract-relax exercise applied with therapeutic massage can reduce LLI and increased lower extremity muscle strength.

• Korean Journal of Mathematics
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• v.16 no.4
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• pp.527-535
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• 2008

Let $I{\in}C^{1,1}$ be a strongly indefinite functional defined on a Hilbert space H. We investigate the number of the critical points of I when I satisfies one pair of Torus-Sphere variational linking inequality. We show that I has at least two critical points when I satisfies one pair of Torus-Sphere variational linking inequality with $(P.S.)^*_c$ condition. We prove this result by use of the limit relative category and critical point theory on the manifold with boundary.

• Honam Mathematical Journal
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• v.32 no.4
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• pp.537-544
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• 2010

We prove the superstability of a functional inequality associated with general exponential functions as follows; ${\mid}f(x+y)-a^{x^2y+xy^2}g(x)f(y){\mid}{\leq}H_p(x,y)$. It is a generalization of the superstability theorem for the exponential functional equation proved by Baker.

• Nonlinear Functional Analysis and Applications
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• v.27 no.4
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• pp.797-805
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• 2022

Let P(z) be a polynomial of degree n. A well-known inequality due to S. Bernstein states that if P ∈ P_n, then $$\max_{{\mid}z{\mid}=1}\,{\mid}P^{\prime}(z){\mid}\,{\leq}n\,\max_{{\mid}z{\mid}=1}\,{\mid}P(z){\mid}$$. In this paper, we establish some extensions and refinements of the above inequality to polar derivative and some other well-known inequalities concerning the polynomials and their ordinary derivatives.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.12 no.4
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• pp.239-247
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• 2008

Let $I{\in}C^{1,1}$ be a strongly indefinite functional defined on a Hilbert space H. We investigate the number of the critical points of I when I satisfies two pairs of Torus-Sphere variational linking inequalities and when I satisfies m ($m{\geq}2$) pairs of Torus-Sphere variational linking inequalities. We show that I has at least four critical points when I satisfies two pairs of Torus-Sphere variational linking inequality with $(P.S.)^*_c$ condition. Moreover we show that I has at least 2m critical points when I satisfies m ($m{\geq}2$) pairs of Torus-Sphere variational linking inequalities with $(P.S.)^*_c$ condition. We prove these results by Theorem 2.2 (Theorem 1.1 in [1]) and the critical point theory on the manifold with boundary.

• Journal of the Korean Mathematical Society
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• v.53 no.1
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• pp.57-72
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• 2016

We establish maximal moment inequalities of partial sums under ${\psi}$-weak dependence, which has been proposed by Doukhan and Louhichi [P. Doukhan and S. Louhichi, A new weak dependence condition and application to moment inequality, Stochastic Process. Appl. 84 (1999), 313-342], to unify weak dependence such as mixing, association, Gaussian sequences and Bernoulli shifts. As an application of maximal moment inequalities, a functional central limit theorem is developed for linear processes with ${\psi}$-weakly dependent innovations.