Journal of Inequalities and Applications最新文献

{"title":"Differential equation and inequalities of the generalized k-Bessel functions.","authors":"Saiful R Mondal,&nbsp;Mohamed S Akel","doi":"10.1186/s13660-018-1772-1","DOIUrl":"https://doi.org/10.1186/s13660-018-1772-1","url":null,"abstract":"<p><p>In this paper, we introduce and study a generalization of the k-Bessel function of order <i>ν</i> given by <dispformula><math><msubsup><mi>W</mi><mrow><mi>ν</mi><mo>,</mo><mi>c</mi></mrow><mi>k</mi></msubsup><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow><mi>∞</mi></munderover><mfrac><msup><mrow><mo>(</mo><mo>-</mo><mi>c</mi><mo>)</mo></mrow><mi>r</mi></msup><mrow><msub><mi>Γ</mi><mi>k</mi></msub><mo>(</mo><mi>r</mi><mi>k</mi><mo>+</mo><mi>ν</mi><mo>+</mo><mi>k</mi><mo>)</mo><mi>r</mi><mo>!</mo></mrow></mfrac><msup><mrow><mo>(</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>)</mo></mrow><mrow><mn>2</mn><mi>r</mi><mo>+</mo><mfrac><mi>ν</mi><mi>k</mi></mfrac></mrow></msup><mo>.</mo></math></dispformula> We also indicate some representation formulae for the function introduced. Further, we show that the function <math><msubsup><mi>W</mi><mrow><mi>ν</mi><mo>,</mo><mi>c</mi></mrow><mi>k</mi></msubsup></math> is a solution of a second-order differential equation. We investigate monotonicity and log-convexity properties of the generalized k-Bessel function <math><msubsup><mi>W</mi><mrow><mi>ν</mi><mo>,</mo><mi>c</mi></mrow><mi>k</mi></msubsup></math> , particularly, in the case <math><mi>c</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> . We establish several inequalities, including a Turán-type inequality. We propose an open problem regarding the pattern of the zeroes of <math><msubsup><mi>W</mi><mrow><mi>ν</mi><mo>,</mo><mi>c</mi></mrow><mi>k</mi></msubsup></math> .</p>","PeriodicalId":49163,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/s13660-018-1772-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36421825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Approximating distance between sets by multivalued coupling with application to uniformly convex Banach spaces.","authors":"Binayak S Choudhury,&nbsp;Pranati Maity,&nbsp;Nikhilesh Metiya,&nbsp;Mihai Postolache","doi":"10.1186/s13660-018-1720-0","DOIUrl":"https://doi.org/10.1186/s13660-018-1720-0","url":null,"abstract":"<p><p>In this paper, our aim is to ascertain the distance between two sets iteratively in two simultaneous ways with the help of a multivalued coupling define for this purpose. We define the best proximity points of such couplings that realize the distance between two sets. Our main theorem is deduced in metric spaces. As an application, we obtain the corresponding results in uniformly convex Banach spaces using the geometry of the space. We discuss two examples.</p>","PeriodicalId":49163,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/s13660-018-1720-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36422927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Smoothing approximation to the lower order exact penalty function for inequality constrained optimization.","authors":"Shujun Lian,&nbsp;Nana Niu","doi":"10.1186/s13660-018-1723-x","DOIUrl":"https://doi.org/10.1186/s13660-018-1723-x","url":null,"abstract":"<p><p>For inequality constrained optimization problem, we first propose a new smoothing method to the lower order exact penalty function, and then show that an approximate global solution of the original problem can be obtained by solving a global solution of a smooth lower order exact penalty problem. We propose an algorithm based on the smoothed lower order exact penalty function. The global convergence of the algorithm is proved under some mild conditions. Some numerical experiments show the efficiency of the proposed method.</p>","PeriodicalId":49163,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/s13660-018-1723-x","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36422929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On a class of N-dimensional anisotropic Sobolev inequalities.","authors":"Lirong Huang,&nbsp;Eugenio Rocha","doi":"10.1186/s13660-018-1754-3","DOIUrl":"https://doi.org/10.1186/s13660-018-1754-3","url":null,"abstract":"<p><p>In this paper, we study the smallest constant <i>α</i> in the anisotropic Sobolev inequality of the form <dispformula><math><msubsup><mrow><mo>∥</mo><mi>u</mi><mo>∥</mo></mrow><mi>p</mi><mi>p</mi></msubsup><mo>≤</mo><mi>α</mi><msubsup><mrow><mo>∥</mo><mi>u</mi><mo>∥</mo></mrow><mn>2</mn><mfrac><mrow><mn>2</mn><mo>(</mo><mn>2</mn><mi>N</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>+</mo><mo>(</mo><mn>3</mn><mo>-</mo><mn>2</mn><mi>N</mi><mo>)</mo><mi>p</mi></mrow><mn>2</mn></mfrac></msubsup><msubsup><mrow><mo>∥</mo><msub><mi>u</mi><mi>x</mi></msub><mo>∥</mo></mrow><mn>2</mn><mfrac><mrow><mi>N</mi><mo>(</mo><mi>p</mi><mo>-</mo><mn>2</mn><mo>)</mo></mrow><mn>2</mn></mfrac></msubsup><munderover><mo>∏</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></munderover><msubsup><mrow><mo>∥</mo><msubsup><mi>D</mi><mi>x</mi><mrow><mo>-</mo><mn>1</mn></mrow></msubsup><msub><mi>∂</mi><msub><mi>y</mi><mi>k</mi></msub></msub><mi>u</mi><mo>∥</mo></mrow><mn>2</mn><mfrac><mrow><mi>p</mi><mo>-</mo><mn>2</mn></mrow><mn>2</mn></mfrac></msubsup></math></dispformula> and the smallest constant <i>β</i> in the inequality <dispformula><math><msubsup><mrow><mo>∥</mo><mi>u</mi><mo>∥</mo></mrow><msub><mi>p</mi><mo>∗</mo></msub><msub><mi>p</mi><mo>∗</mo></msub></msubsup><mo>≤</mo><mi>β</mi><msubsup><mrow><mo>∥</mo><msub><mi>u</mi><mi>x</mi></msub><mo>∥</mo></mrow><mn>2</mn><mfrac><mrow><mn>2</mn><mi>N</mi></mrow><mrow><mn>2</mn><mi>N</mi><mo>-</mo><mn>3</mn></mrow></mfrac></msubsup><munderover><mo>∏</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></munderover><msubsup><mrow><mo>∥</mo><msubsup><mi>D</mi><mi>x</mi><mrow><mo>-</mo><mn>1</mn></mrow></msubsup><msub><mi>∂</mi><msub><mi>y</mi><mi>k</mi></msub></msub><mi>u</mi><mo>∥</mo></mrow><mn>2</mn><mfrac><mn>2</mn><mrow><mn>2</mn><mi>N</mi><mo>-</mo><mn>3</mn></mrow></mfrac></msubsup><mo>,</mo></math></dispformula> where <math><mi>V</mi><mo>:</mo><mo>=</mo><mo>(</mo><mi>x</mi><mo>,</mo><msub><mi>y</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>y</mi><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>∈</mo><msup><mi>R</mi><mi>N</mi></msup></math> with <math><mi>N</mi><mo>≥</mo><mn>3</mn></math> and <math><mn>2</mn><mo><</mo><mi>p</mi><mo><</mo><msub><mi>p</mi><mo>∗</mo></msub><mo>=</mo><mfrac><mrow><mn>2</mn><mo>(</mo><mn>2</mn><mi>N</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn><mi>N</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math> . These constants are characterized by variational methods and scaling techniques. The techniques used here seem to have independent interests.</p>","PeriodicalId":49163,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/s13660-018-1754-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36422939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Spectral properties of an impulsive Sturm-Liouville operator.","authors":"Elgiz Bairamov,&nbsp;Ibrahim Erdal,&nbsp;Seyhmus Yardimci","doi":"10.1186/s13660-018-1781-0","DOIUrl":"https://doi.org/10.1186/s13660-018-1781-0","url":null,"abstract":"<p><p>This work is devoted to discuss some spectral properties and the scattering function of the impulsive operator generated by the Sturm-Liouville equation. We present a different method to investigate the spectral singularities and eigenvalues of the mentioned operator. We also obtain the finiteness of eigenvalues and spectral singularities with finite multiplicities under some certain conditions. Finally, we illustrate our results by a detailed example.</p>","PeriodicalId":49163,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/s13660-018-1781-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36419180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sobolev type inequalities for compact metric graphs.","authors":"Muhammad Usman","doi":"10.1186/s13660-018-1872-y","DOIUrl":"https://doi.org/10.1186/s13660-018-1872-y","url":null,"abstract":"<p><p>In this paper analogues of Sobolev inequalities for compact and connected metric graphs are derived. As a consequence of these inequalities, a lower bound, commonly known as Cheeger inequality, on the first non-zero eigenvalue of the Laplace operator with standard vertex conditions is recovered.</p>","PeriodicalId":49163,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/s13660-018-1872-y","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36664774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Gradient estimates and Liouville-type theorems for a weighted nonlinear elliptic equation.","authors":"Bingqing Ma,&nbsp;Yongli Dong","doi":"10.1186/s13660-018-1705-z","DOIUrl":"https://doi.org/10.1186/s13660-018-1705-z","url":null,"abstract":"<p><p>We consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a smooth metric measure space [Formula: see text]: [Formula: see text] where <i>a</i>, <i>b</i> are two real constants. When the ∞-Bakry-Émery Ricci curvature is bounded from below, we obtain a global gradient estimate which is not dependent on [Formula: see text]. In particular, we find that any bounded positive solution of the above equation must be constant under some suitable assumptions.</p>","PeriodicalId":49163,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/s13660-018-1705-z","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36109942","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An extended reverse Hardy-Hilbert's inequality in the whole plane.","authors":"Qiang Chen,&nbsp;Bicheng Yang","doi":"10.1186/s13660-018-1706-y","DOIUrl":"https://doi.org/10.1186/s13660-018-1706-y","url":null,"abstract":"<p><p>Using weight coefficients, a complex integral formula, and Hermite-Hadamard's inequality, we give an extended reverse Hardy-Hilbert's inequality in the whole plane with multiparameters and a best possible constant factor. Equivalent forms and a few particular cases are considered.</p>","PeriodicalId":49163,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/s13660-018-1706-y","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36114853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Gradient projection method with a new step size for the split feasibility problem.","authors":"Pattanapong Tianchai","doi":"10.1186/s13660-018-1712-0","DOIUrl":"10.1186/s13660-018-1712-0","url":null,"abstract":"<p><p>In this paper, we introduce an iterative scheme using the gradient projection method with a new step size, which is not depend on the related matrix inverses and the largest eigenvalue (or the spectral radius of the self-adjoint operator) of the related matrix, based on Moudafi's viscosity approximation method for solving the <i>split feasibility problem</i> (SFP), which is to find a point in a given closed convex subset of a real Hilbert space such that its image under a bounded linear operator belongs to a given closed convex subset of another real Hilbert space. We suggest and analyze this iterative scheme under some appropriate conditions imposed on the parameters such that another strong convergence theorems for the SFP are obtained. The results presented in this paper improve and extend the main results of Tian and Zhang (J. Inequal. Appl. 2017:Article ID 13, 2017), and Tang et al. (Acta Math. Sci. 36B(2):602-613, 2016) (in a single-step regularized method) with a new step size, and many others. The examples of the proposed SFP are also shown through numerical results.</p>","PeriodicalId":49163,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/s13660-018-1712-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36134898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"New bounds for the exponential function with cotangent.","authors":"Ling Zhu","doi":"10.1186/s13660-018-1697-8","DOIUrl":"10.1186/s13660-018-1697-8","url":null,"abstract":"<p><p>In this paper, new bounds for the exponential function with cotangent are found by using the recurrence relation between coefficients in the expansion of power series of the function [Formula: see text] and a new criterion for the monotonicity of the quotient of two power series.</p>","PeriodicalId":49163,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5940775/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36105458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}