Pub Date : 2024-09-03DOI: 10.1186/s13660-024-03192-4
M. Ashraf Bhat, G. Sankara Raju Kosuru
We establish trace inequalities for Riesz potentials on Herz-type spaces and examine the optimality of conditions imposed on specific parameters. We also present some applications in the form of Sobolev-type inequalities, including the Gagliardo–Nirenberg–Sobolev inequality and the fractional integration theorem in the Herz space setting. In addition, we obtain a Sobolev embedding theorem for Herz-type Sobolev spaces.
{"title":"Trace principle for Riesz potentials on Herz-type spaces and applications","authors":"M. Ashraf Bhat, G. Sankara Raju Kosuru","doi":"10.1186/s13660-024-03192-4","DOIUrl":"https://doi.org/10.1186/s13660-024-03192-4","url":null,"abstract":"We establish trace inequalities for Riesz potentials on Herz-type spaces and examine the optimality of conditions imposed on specific parameters. We also present some applications in the form of Sobolev-type inequalities, including the Gagliardo–Nirenberg–Sobolev inequality and the fractional integration theorem in the Herz space setting. In addition, we obtain a Sobolev embedding theorem for Herz-type Sobolev spaces.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-02DOI: 10.1186/s13660-024-03189-z
Slavica Ivelić Bradanović, Neda Lovričević
Strongly convex functions as a subclass of convex functions, still equipped with stronger properties, are employed through several generalizations and improvements of the Jensen inequality and the Jensen–Mercer inequality. This paper additionally provides applications of obtained main results in the form of new estimates for so-called strong f-divergences: the concept of the Csiszár f-divergence for strongly convex functions f, together with particular cases (Kullback–Leibler divergence, $chi ^{2}$ -divergence, Hellinger divergence, Bhattacharya distance, Jeffreys distance, and Jensen–Shannon divergence.) Furthermore, new estimates for the Shannon entropy are obtained, and new Chebyshev-type inequalities are derived.
强凸函数作为凸函数的一个子类，仍然具有更强的性质，通过对詹森不等式和詹森-默塞尔不等式的几种概括和改进而得到应用。本文还以所谓强 f 发散的新估计的形式提供了所获主要结果的应用：强凸函数 f 的 Csiszár f 发散概念以及特殊情况（Kullback-Leibler 发散、$chi ^{2}$ -发散、Hellinger 发散、Bhattacharya 距离、Jeffreys 距离和 Jensen-Shannon 发散）。此外，还得到了香农熵的新估计值，并推导出新的切比雪夫型不等式。
{"title":"Generalized Jensen and Jensen–Mercer inequalities for strongly convex functions with applications","authors":"Slavica Ivelić Bradanović, Neda Lovričević","doi":"10.1186/s13660-024-03189-z","DOIUrl":"https://doi.org/10.1186/s13660-024-03189-z","url":null,"abstract":"Strongly convex functions as a subclass of convex functions, still equipped with stronger properties, are employed through several generalizations and improvements of the Jensen inequality and the Jensen–Mercer inequality. This paper additionally provides applications of obtained main results in the form of new estimates for so-called strong f-divergences: the concept of the Csiszár f-divergence for strongly convex functions f, together with particular cases (Kullback–Leibler divergence, $chi ^{2}$ -divergence, Hellinger divergence, Bhattacharya distance, Jeffreys distance, and Jensen–Shannon divergence.) Furthermore, new estimates for the Shannon entropy are obtained, and new Chebyshev-type inequalities are derived.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-30DOI: 10.1186/s13660-024-03179-1
Sundas Nawaz, Murad Khan Hassani, Afshan Batool, Ali Akgül
In the present article, the Hyers–Ulam stability of the following inequality is analyzed: 0.1 $$ textstylebegin{cases} d (f(imath +jmath ), (f(imath )+ f(jmath )) )leq d (rho _{1}((f(imath +jmath )+ f(imath - jmath ), 2f(imath )) ) hphantom{ d (f(imath +jmath ), (f(imath )+ f(jmath )) )leq}{}+ d (rho _{2} (2f (frac{imath +jmath}{2} ), (f(imath )+ f(jmath )) ) ) end{cases} $$ in the setting of digital metric space, where $rho _{1}$ and $rho _{2}$ are fixed nonzero complex numbers with $1>sqrt{2}|rho _{1}|+|rho _{2}|$ by using fixed point and direct approach.
{"title":"Stability of functional inequality in digital metric space","authors":"Sundas Nawaz, Murad Khan Hassani, Afshan Batool, Ali Akgül","doi":"10.1186/s13660-024-03179-1","DOIUrl":"https://doi.org/10.1186/s13660-024-03179-1","url":null,"abstract":"In the present article, the Hyers–Ulam stability of the following inequality is analyzed: 0.1 $$ textstylebegin{cases} d (f(imath +jmath ), (f(imath )+ f(jmath )) )leq d (rho _{1}((f(imath +jmath )+ f(imath - jmath ), 2f(imath )) ) hphantom{ d (f(imath +jmath ), (f(imath )+ f(jmath )) )leq}{}+ d (rho _{2} (2f (frac{imath +jmath}{2} ), (f(imath )+ f(jmath )) ) ) end{cases} $$ in the setting of digital metric space, where $rho _{1}$ and $rho _{2}$ are fixed nonzero complex numbers with $1>sqrt{2}|rho _{1}|+|rho _{2}|$ by using fixed point and direct approach.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209593","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose two hybrid methods for solving large-scale monotone systems, which are based on derivative-free conjugate gradient approach and hyperplane projection technique. The conjugate gradient approach is efficient for large-scale systems due to low memory, while projection strategy is suitable for monotone equations because it enables simply globalization. The derivative-free function-value-based line search is combined with Hu-Storey type search directions and projection procedure, in order to construct globally convergent methods. Furthermore, the proposed methods are applied into solving a number of large-scale monotone nonlinear systems and reconstruction of sparse signals. Numerical experiments indicate the robustness of the proposed methods.
{"title":"Hybrid Hu-Storey type methods for large-scale nonlinear monotone systems and signal recovery","authors":"Zoltan Papp, Sanja Rapajić, Abdulkarim Hassan Ibrahim, Supak Phiangsungnoen","doi":"10.1186/s13660-024-03187-1","DOIUrl":"https://doi.org/10.1186/s13660-024-03187-1","url":null,"abstract":"We propose two hybrid methods for solving large-scale monotone systems, which are based on derivative-free conjugate gradient approach and hyperplane projection technique. The conjugate gradient approach is efficient for large-scale systems due to low memory, while projection strategy is suitable for monotone equations because it enables simply globalization. The derivative-free function-value-based line search is combined with Hu-Storey type search directions and projection procedure, in order to construct globally convergent methods. Furthermore, the proposed methods are applied into solving a number of large-scale monotone nonlinear systems and reconstruction of sparse signals. Numerical experiments indicate the robustness of the proposed methods.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper offers a comprehensive analysis of solution representations for ϖ-fractional partial differential equations, specifically focusing on the linear case of the Darboux problem. We exhibit a representation of the solutions for the Darboux problem of ϖ-fractional partial differential equations in the linear case in the space of continuous functions. Through the application of the generalized Gronwall inequality, we establish the Ulam–Hyers–Rassias Mittag–Leffler stability in the space of continuous functions. Three numerical examples are presented to show the effectiveness and the applicability of our results.
{"title":"Ulam–Hyers–Rassias Mittag-Leffler stability of ϖ–fractional partial differential equations","authors":"Mohamed Rhaima, Djalal Boucenna, Lassaad Mchiri, Mondher Benjemaa, Abdellatif Ben Makhlouf","doi":"10.1186/s13660-024-03170-w","DOIUrl":"https://doi.org/10.1186/s13660-024-03170-w","url":null,"abstract":"This paper offers a comprehensive analysis of solution representations for ϖ-fractional partial differential equations, specifically focusing on the linear case of the Darboux problem. We exhibit a representation of the solutions for the Darboux problem of ϖ-fractional partial differential equations in the linear case in the space of continuous functions. Through the application of the generalized Gronwall inequality, we establish the Ulam–Hyers–Rassias Mittag–Leffler stability in the space of continuous functions. Three numerical examples are presented to show the effectiveness and the applicability of our results.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-27DOI: 10.1186/s13660-024-03186-2
Stevo Stević, Bratislav Iričanin, Witold Kosmala, Zdeněk Šmarda
We show that the system of difference equations $$ x_{n+k}=frac{x_{n+l}y_{n}-ef}{x_{n+l}+y_{n}-e-f},quad y_{n+k}= frac{y_{n+l}x_{n}-ef}{y_{n+l}+x_{n}-e-f},quad nin {mathbb{N}}_{0}, $$ where $kin {mathbb{N}}$ , $lin {mathbb{N}}_{0}$ , $l< k$ , $e, fin {mathbb{C}}$ , and $x_{j}, y_{j}in {mathbb{C}}$ , $j=overline{0,k-1}$ , is theoretically solvable and present some cases of the system when the general solutions can be found in a closed form.
{"title":"On solvability of a two-dimensional symmetric nonlinear system of difference equations","authors":"Stevo Stević, Bratislav Iričanin, Witold Kosmala, Zdeněk Šmarda","doi":"10.1186/s13660-024-03186-2","DOIUrl":"https://doi.org/10.1186/s13660-024-03186-2","url":null,"abstract":"We show that the system of difference equations $$ x_{n+k}=frac{x_{n+l}y_{n}-ef}{x_{n+l}+y_{n}-e-f},quad y_{n+k}= frac{y_{n+l}x_{n}-ef}{y_{n+l}+x_{n}-e-f},quad nin {mathbb{N}}_{0}, $$ where $kin {mathbb{N}}$ , $lin {mathbb{N}}_{0}$ , $l< k$ , $e, fin {mathbb{C}}$ , and $x_{j}, y_{j}in {mathbb{C}}$ , $j=overline{0,k-1}$ , is theoretically solvable and present some cases of the system when the general solutions can be found in a closed form.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-26DOI: 10.1186/s13660-024-03183-5
Khangembam Babina Devi, Barchand Chanam
Let $p(z)$ be a polynomial of degree n having no zero in $|z|< k$ , $kleq 1$ , then Govil [Proc. Nat. Acad. Sci., 50(1980), 50-52] proved $$ max _{|z|=1}|p'(z)|leq frac{n}{1+k^{n}}max _{|z|=1}|p(z)|, $$ provided $|p'(z)|$ and $|q'(z)|$ attain their maxima at the same point on the circle $|z|=1$ , where $$ q(z)=z^{n}overline{pbigg(frac{1}{overline{z}}bigg)}. $$ In this paper, we present integral mean inequalities of Turán- and Erdös-Lax-type for the polar derivative of a polynomial by involving some coefficients of the polynomial, which refine some previously proved results and one of our results improves the above Govil inequality as a special case. These results incorporate the placement of the zeros and some coefficients of the underlying polynomial. Furthermore, we provide numerical examples and graphical representations to demonstrate the superior precision of our results compared to some previously established results.
{"title":"On Bernstein and Turán-type integral mean estimates for polar derivative of a polynomial","authors":"Khangembam Babina Devi, Barchand Chanam","doi":"10.1186/s13660-024-03183-5","DOIUrl":"https://doi.org/10.1186/s13660-024-03183-5","url":null,"abstract":"Let $p(z)$ be a polynomial of degree n having no zero in $|z|< k$ , $kleq 1$ , then Govil [Proc. Nat. Acad. Sci., 50(1980), 50-52] proved $$ max _{|z|=1}|p'(z)|leq frac{n}{1+k^{n}}max _{|z|=1}|p(z)|, $$ provided $|p'(z)|$ and $|q'(z)|$ attain their maxima at the same point on the circle $|z|=1$ , where $$ q(z)=z^{n}overline{pbigg(frac{1}{overline{z}}bigg)}. $$ In this paper, we present integral mean inequalities of Turán- and Erdös-Lax-type for the polar derivative of a polynomial by involving some coefficients of the polynomial, which refine some previously proved results and one of our results improves the above Govil inequality as a special case. These results incorporate the placement of the zeros and some coefficients of the underlying polynomial. Furthermore, we provide numerical examples and graphical representations to demonstrate the superior precision of our results compared to some previously established results.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209596","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-23DOI: 10.1186/s13660-024-03190-6
Babak Mohammadi, Vahid Parvaneh, Mohammad Mursaleen
In this study, we prove the existence and uniqueness of a solution to a g-Caputo fractional differential equation with new boundary value conditions utilizing the combined Wardowski–Mizoguchi–Takahashi contractions via reduction of this equation to a fractional integral equation. We provide an example to demonstrate our findings.
{"title":"Existence of solution for some nonlinear g-Caputo fractional-order differential equations based on Wardowski–Mizoguchi–Takahashi contractions","authors":"Babak Mohammadi, Vahid Parvaneh, Mohammad Mursaleen","doi":"10.1186/s13660-024-03190-6","DOIUrl":"https://doi.org/10.1186/s13660-024-03190-6","url":null,"abstract":"In this study, we prove the existence and uniqueness of a solution to a g-Caputo fractional differential equation with new boundary value conditions utilizing the combined Wardowski–Mizoguchi–Takahashi contractions via reduction of this equation to a fractional integral equation. We provide an example to demonstrate our findings.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209599","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-23DOI: 10.1186/s13660-024-03184-4
Paul Bosch, José M. Rodríguez, José M. Sigarreta, Eva Tourís
Inequalities play a main role in pure and applied mathematics. In this paper, we prove a generalization of Milne inequality for any measure space. The argument in the proof of this inequality allows us to obtain other Milne-type inequalities. Also, we improve the discrete version of Milne inequality, which holds for any positive value of the parameter p. Finally, we present a Milne-type inequality in the fractional context.
不等式在纯数学和应用数学中发挥着重要作用。本文证明了任何度量空间的米尔恩不等式的广义化。通过证明这个不等式的论证，我们可以得到其他米尔恩型不等式。此外，我们还改进了离散版的米尔恩不等式，该不等式对于参数 p 的任何正值都成立。
{"title":"Some new Milne-type inequalities","authors":"Paul Bosch, José M. Rodríguez, José M. Sigarreta, Eva Tourís","doi":"10.1186/s13660-024-03184-4","DOIUrl":"https://doi.org/10.1186/s13660-024-03184-4","url":null,"abstract":"Inequalities play a main role in pure and applied mathematics. In this paper, we prove a generalization of Milne inequality for any measure space. The argument in the proof of this inequality allows us to obtain other Milne-type inequalities. Also, we improve the discrete version of Milne inequality, which holds for any positive value of the parameter p. Finally, we present a Milne-type inequality in the fractional context.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-14DOI: 10.1186/s13660-024-03185-3
Mouataz Billah Mesmouli, Farah M. Al-Askar, Wael W. Mohammed
In this paper, a $left ( p,qright ) $ -fractional nonlinear difference equation of different orders is considered and discussed. With the help of $left ( p,qright ) $ -calculus for integrals and derivatives properties, we convert the main integral boundary value problem (IBVP) to an equivalent solution in the form of an integral equation, we use the upper–lower solution technique to prove the existence of positive solutions. We present an example of the IBVP to apply and demonstrate the results of our method.
{"title":"Upper and lower solutions for an integral boundary problem with two different orders (left ( p,qright ) )-fractional difference","authors":"Mouataz Billah Mesmouli, Farah M. Al-Askar, Wael W. Mohammed","doi":"10.1186/s13660-024-03185-3","DOIUrl":"https://doi.org/10.1186/s13660-024-03185-3","url":null,"abstract":"In this paper, a $left ( p,qright ) $ -fractional nonlinear difference equation of different orders is considered and discussed. With the help of $left ( p,qright ) $ -calculus for integrals and derivatives properties, we convert the main integral boundary value problem (IBVP) to an equivalent solution in the form of an integral equation, we use the upper–lower solution technique to prove the existence of positive solutions. We present an example of the IBVP to apply and demonstrate the results of our method.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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