• Nonlinear Functional Analysis and Applications
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• v.28 no.4
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• pp.919-928
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• 2023

Let P(z) be a polynomial of degree n and P(z) ≠ 0 in |z| < 1. Then for every real α and R > 1, Aziz [1] proved that $$\max\limits_{{\mid}z{\mid}=1}{\mid}P(Rz)-P(z){\mid}{\leq}{\frac{R^n-1}{2}}(M^2_{\alpha}+M^2_{{\alpha}+{\pi}})^{\frac{1}{2}}{\mid},$$ where $$M{\alpha}={\max\limits_{1{\leq}k{\leq}n}}{\mid}P(e^{i({\alpha}+2k{\pi})n}){\mid}.$$ In this paper, we establish some improvements and generalizations of the above inequality concerning the polynomials and their ordinary derivatives.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.4
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• pp.437-445
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• 2008

We show the existence of at least four nontrivial critical points of the $C^{1,1}$ functional f on the Hilbert space $H=X_0{\oplus}X_1{\oplus}X_2{\oplus}X_3{\oplus}X_4$, $X_i$, i = 0, 1, 2, 3 are finite dimensional, with f(0) = 0 when two sublevel subsets, torus with three holes and sphere, of f link, the functional f satisfies sup-inf variatinal linking inequality on the linking subspaces, the functional f satisfies $(P.S.)_c$ condition, and $f{\mid}_{X_0{\oplus}X_4}$ has no critical point with level c. We use the deformation lemma, the relative category theory and the critical point theory for the proof of main result.

• Nonlinear Functional Analysis and Applications
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• v.27 no.3
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• pp.691-700
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• 2022

Let p(z) be a polynomial of degree n having no zero inside the unit circle. Then for 0 < r ≤ 1, the well-known inequality due to Rivlin [Amer. Math. Monthly., 67 (1960) 251-253] is $$\max\limits_{{\mid}z{\mid}=r}{\mid}p(z){\mid}{\geq}{\(\frac{r+1}{2}\)^n}\max\limits_{{\mid}z{\mid}=1}{\mid}p(z){\mid}$$. In this paper, we generalize as well as sharpen the above inequality. Also our results not only generalize, but also sharpen some known results proved recently.

• Nonlinear Functional Analysis and Applications
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• v.28 no.3
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• pp.731-751
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• 2023

If p(z) is a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, then for any complex number α with |α| ≥ k, and r ≥ 1, Aziz [1] proved $$\left{{\int}_{0}^{2{\pi}}\,{\left|1+k^ne^{i{\theta}}\right|^r}\,d{\theta}\right}^{\frac{1}{r}}\;{\max\limits_{{\mid}z{\mid}=1}}\,{\mid}p^{\prime}(z){\mid}\,{\geq}\,n\,\left{{\int}_{0}^{2{\pi}}\,{\left|p(e^{i{\theta}})\right|^r\,d{\theta}\right}^{\frac{1}{r}}.$$ In this paper, we obtain an improved extension of the above inequality into polar derivative. Further, we also extend an inequality on polar derivative recently proved by Rather et al. [20] into L^r-norm. Our results not only extend some known polynomial inequalities, but also reduce to some interesting results as particular cases.

• Annals of Rehabilitation Medicine
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• v.42 no.6
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• pp.863-871
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• 2018

Objective To evaluate the association between progression of curvature of scoliosis, and correction for functional component in patients with juvenile idiopathic scoliosis (JIS). Methods We retrospectively reviewed medical data of patients prescribed custom molded foot orthosis (FO) to correct inequality of RCSPA (resting calcaneal stance position angle), and chose 52 patients (26 females, 26 males) with Cobb angle ${\geq}10^{\circ}$ in radiology and uneven pelvic level at iliac crest by different RCSPA (${\geq}3^{\circ}$) as a factor of functional scoliosis. They had different hump angle ${\geq}5^{\circ}$ in forward bending test, for idiopathic scoliosis component. Their mean age and mean period of wearing FO were $79.5{\pm}10.6months$ and $18.6{\pm}0.70months$. Results Cobb angle was reduced from $22.03^{\circ}{\pm}4.39^{\circ}$ initially to $18.86^{\circ}{\pm}7.53^{\circ}$ after wearing FO. Pelvis height difference and RCSPA difference, were reduced from $1.07{\pm}0.25cm$ initially to $0.60{\pm}0.36$, and from $4.25^{\circ}{\pm}0.71^{\circ}$ initially to $1.71^{\circ}{\pm}0.75^{\circ}$ (p<0.01). Cobb angle improved most in 9 months. However, there was no significant improvement for those with more than $25^{\circ}$ of Cobb angle initially. Mean Cobb angle improved in all age groups, but patients less than 6 years had clinically significant improvement of more than $5^{\circ}$. Conclusion JIS can have functional components, which should be identified and managed. Foot orthosis is useful in correcting functional factors, in the case of pelvic inequality caused by different RCSPA, for patients with juvenile idiopathic scoliosis.

• Nonlinear Functional Analysis and Applications
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• v.28 no.1
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• pp.205-235
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• 2023

In this paper, we introduce and study an iterative technique for solving quasimonotone split variational inequality problems and fixed point problem in the framework of real Hilbert spaces. Our proposed iterative technique is self adaptive, and easy to implement. We establish that the proposed iterative technique converges strongly to a minimum-norm solution of the problem and give some numerical illustrations in comparison with other methods in the literature to support our strong convergence result.

• Bulletin of the Korean Mathematical Society
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• v.46 no.2
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• pp.311-319
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• 2009

Let H be a Hilbert space which is the direct sum of five closed subspaces $X_0,\;X_1,\;X_2,\;X_3$ and $X_4$ with $X_1,\;X_2,\;X_3$ of finite dimension. Let J be a $C^{1,1}$ functional defined on H with J(0) = 0. We show the existence of at least four nontrivial critical points when the sublevels of J (the torus with three holes and sphere) link and the functional J satisfies sup-inf variational inequality on the linking subspaces, and the functional J satisfies $(P.S.)^*_c$ condition and $f|X_0{\otimes}X_4$ has no critical point with level c. For the proof of main theorem we use the nonsmooth version of the classical deformation lemma and the limit relative category theory.

• Journal of the Korean Society of Physical Medicine
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• v.2 no.2
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• pp.101-112
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• 2007

Purpose : The purpose of this study was to evaluate influence of sacroiliac joint mobilization on lower extremity muscle strength. Methods : The subjects were consisted of thirty patients who had Leg length inequality(LLI) of more than 10mm(16 females. 14 males) from 21 to 41 years of age(mean aged 24.87). All subjects randomly assigned to sacroiliac joint mobilization group(n=15), control group(n=15). sacroiliac joint mobilization group received sacroiliac joint mobilization about 10 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 sacroiliac joint mobilization group was significantly reduced according to within treatment period(p<.05), most significantly reduced between pre-test and post-test(p<.05). sacroiliac joint mobilization group significantly more reduced than control group(p<.05). 2. The knee extension strength of sacroiliac joint mobilization group was significantly increased according to within treatment period(p<.05), most significantly increased between pre-test and post-test(p<.05). sacroiliac joint mobilization group significantly more increased than control group(p<.05). 3. The knee flexion strength of sacroiliac joint mobilization group was significantly increased according to within treatment period(p<.05), most significantly increased between pre-test and post-test(p<.05). sacroiliac joint mobilization group significantly more increased than control group(p<.05). Conclusion : sacroiliac joint mobilization can reduce LLI and increased lower extremity muscle strength.

• Honam Mathematical Journal
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• v.31 no.1
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• pp.75-85
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• 2009

We show the existence of the nontrivial periodic solution for a class of the system of the nonlinear suspension bridge equations with Dirichlet boundary condition and periodic condition by critical point theory and linking arguments. We investigate the geometry of the sublevel sets of the corresponding functional of the system, the topology of the sublevel sets and linking construction between two sublevel sets. Since the functional is strongly indefinite, we use the linking theorem for the strongly indefinite functional and the notion of the suitable version of the Palais-Smale condition.

• Bulletin of the Korean Mathematical Society
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• v.47 no.4
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• pp.777-785
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• 2010

Let X be a linear space and Y be a complete quasi p-norm space. We will show that for each function f : X $\rightarrow$ Y, which satisfies the inequality ${\parallel}{\Delta}_x^nf(y)\;-\;n!f(x){\parallel}\;{\leq}\;\varphi(x,y)$ for suitable control function $\varphi$, there is a unique monomial function M of degree n which is a good approximation for f in such a way that the continuity of $t\;{\mapsto}\;f(tx)$ and $t\;{\mapsto}\;\varphi(tx,\;ty)$ imply the continuity of $t\;{\mapsto}\;M(tx)$.