In this research work, we consider the below inequalities: (1.1). The researchers attempt to find an answer as to what are the best possible parameters <i><i>α</i></i>, <i><i&...In this research work, we consider the below inequalities: (1.1). The researchers attempt to find an answer as to what are the best possible parameters <i><i>α</i></i>, <i><i>β</i></i> that (1.1) can be held? The main tool is the optimization of some suitable functions that we seek to find out. Without loss of generality, we have assumed that <i>a</i> > <i>b</i> and let <img src="Edit_26c0f99b-93dd-48ff-acdb-f1c8047744f1.bmp" alt="" /> for 1) and <i>a</i> < <i>b</i>, <img src="Edit_15c32a7a-e9ae-41d3-8f49-c6b9c01c7ece.bmp" alt="" />(<i>t</i> small) for 2) to determine the condition for <i><i>α</i></i> and <i><i>β</i></i> to become <i>f</i>(<i>t</i>) ≤ 0 and <i>g</i>(<i>t</i>) ≥ 0.展开更多
Recently Guo introduced integrated Meyer -Konig and Zeller operators and studied the rate of convergence for function of bounded variation. In this note we give a sharp estimate for these operators.
We introduce a primitive class of analytic functions, by specializing in many wellknown classes, classify Ma-Minda functions based on its conditions and their interesting geometrical aspects. Further, study a newly de...We introduce a primitive class of analytic functions, by specializing in many wellknown classes, classify Ma-Minda functions based on its conditions and their interesting geometrical aspects. Further, study a newly defined subclass of starlike functions involving a special type of Ma-Minda function introduced here for obtaining inclusion and radius results. We also establish some majorization, Bloch function norms, and other related problems for the same class.展开更多
Let B be the unit ball in a complex Banach space. Let S^＊k＋1（B） be the family of normalized starlike mappings f on B such that z = 0 is a zero of order k ＋ 1 of f（z） - z. The authors obtain sharp growth and cov...Let B be the unit ball in a complex Banach space. Let S^＊k＋1（B） be the family of normalized starlike mappings f on B such that z = 0 is a zero of order k ＋ 1 of f（z） - z. The authors obtain sharp growth and covering theorems, as well as sharp coefficient bounds for various subsets of S^＊k＋1（B）.展开更多
We obtain the operator norms of the n-dimensional fractional Hardy operator Hα（0 〈 α 〈 n） from weighted Lebesgue spaces Lp｜x｜p（R^n） to weighted weak Lebesgue spacesLq,∞｜x｜β（R^n）.
Let K be the familiar class of normalized convex functions in the unit disk.In[14],Keogh and Merkes proved that for a function f(z)=z+∑k=2∞a k z k in the class K,|a 3−λa 22|≤max{13,|λ−1|},λ∈C.The above estimate...Let K be the familiar class of normalized convex functions in the unit disk.In[14],Keogh and Merkes proved that for a function f(z)=z+∑k=2∞a k z k in the class K,|a 3−λa 22|≤max{13,|λ−1|},λ∈C.The above estimate is sharp for eachλ.In this article,we establish the corresponding inequality for a normalized convex function f on U such that z=0 is a zero of order k+1 of f(z)−z,and then we extend this result to higher dimensions.These results generalize some known results.展开更多
Let S~* be the familiar class of normalized univalent functions in the unit disk.In [9], Keogh and Merkes proved that for a function f(z) = z +∑k=2∞ a_kz^k in the class S~*,then |a_3-λa_2~2| ≤ max{1, |3-4λ|}, λ...Let S~* be the familiar class of normalized univalent functions in the unit disk.In [9], Keogh and Merkes proved that for a function f(z) = z +∑k=2∞ a_kz^k in the class S~*,then |a_3-λa_2~2| ≤ max{1, |3-4λ|}, λ∈ C. In this article, we investigate the corresponding problem for the subclass of starlike mappings defined on the unit ball in a complex Banach space, on the unit polydisk in Cnand the bounded starlike circular domain in C~■, respectively.展开更多
It was conjectured by Escobar(J Funct Anal 165:101–116,1999)that for an ndimensional(n≥3)smooth compact Riemannian manifold with boundary,which has nonnegative Ricci curvature and boundary principal curvatures bound...It was conjectured by Escobar(J Funct Anal 165:101–116,1999)that for an ndimensional(n≥3)smooth compact Riemannian manifold with boundary,which has nonnegative Ricci curvature and boundary principal curvatures bounded below by c>0,the first nonzero Steklov eigenvalue is greater than or equal to c with equality holding only on isometrically Euclidean balls with radius 1/c.In this paper,we confirm this conjecture in the case of nonnegative sectional curvature.The proof is based on a combination of Qiu-Xia’s weighted Reilly-type formula with a special choice of the weight function depending on the distance function to the boundary,as well as a generalized Pohozaev-type identity.展开更多
In this paper,we calculate the sharp bound for the generalized m-linear n-dimensional Hardy-Littlewood-Polya operator on power weighted central and non-central homogeneous Morrey spaces.As an application,the sharp bou...In this paper,we calculate the sharp bound for the generalized m-linear n-dimensional Hardy-Littlewood-Polya operator on power weighted central and non-central homogeneous Morrey spaces.As an application,the sharp bound for the Hardy-Littlewood-Polya operator on power weighted central and noncentral homo-geneous Morrey spaces is obtained.Finally,we also find the sharp bound for the Hausdorff operator on power weighted central and noncentral homogeneous Morrey spaces,which generalizes the previous results.展开更多
In this paper, we study two different extensions of the Hausdorff operator to the multilinear case. Boundedness on Lebesgue spaces and Herz spaces is obtained. The bound on the Lebesgue space is optimal. Our results a...In this paper, we study two different extensions of the Hausdorff operator to the multilinear case. Boundedness on Lebesgue spaces and Herz spaces is obtained. The bound on the Lebesgue space is optimal. Our results are substantial extensions of some known results on Multilinear high dimensional Hardy operator.展开更多
We offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least n+1/n−1 provided the graph is not complete and that equality is attained if and onl...We offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least n+1/n−1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size n−1/2.With the same method,we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree,provided this is at most n−1/2.展开更多
In this paper,the sharp bound for the weak-type(1,1) inequality for the n-dimensional Hardy operator is obtained.Moreover,the precise norms of generalized Hardy operators on the type of Campanato spaces are worked out...In this paper,the sharp bound for the weak-type(1,1) inequality for the n-dimensional Hardy operator is obtained.Moreover,the precise norms of generalized Hardy operators on the type of Campanato spaces are worked out.As applications,the corresponding norms of the Riemann-Liouville integral operator and the n-dimensional Hardy operator are deduced.It is also proved that the n-dimensional Hardy operator maps from the Hardy space into the Lebesgue space.The endpoint estimate for the commutator generated by the Hardy operator and the(central) BMO function is also discussed.展开更多
In this paper, we establish the Fekete and Szego inequality for a class of holomorphic functions in the unit disk, and then we extend this result to a class of holomorphic mappings on the unit ball in a complex Banach...In this paper, we establish the Fekete and Szego inequality for a class of holomorphic functions in the unit disk, and then we extend this result to a class of holomorphic mappings on the unit ball in a complex Banach space or on the unit polydisk in Cn.展开更多
Let Sα*be the familiar class of normalized starlike functions of order α in the unit disk. In this paper, we establish the Fekete and Szeg? inequality for the class Sα*, and then we generalize this result to the un...Let Sα*be the familiar class of normalized starlike functions of order α in the unit disk. In this paper, we establish the Fekete and Szeg? inequality for the class Sα*, and then we generalize this result to the unit ball in a complex Banach space or on the unit polydisk in Cn.展开更多
文摘In this research work, we consider the below inequalities: (1.1). The researchers attempt to find an answer as to what are the best possible parameters <i><i>α</i></i>, <i><i>β</i></i> that (1.1) can be held? The main tool is the optimization of some suitable functions that we seek to find out. Without loss of generality, we have assumed that <i>a</i> > <i>b</i> and let <img src="Edit_26c0f99b-93dd-48ff-acdb-f1c8047744f1.bmp" alt="" /> for 1) and <i>a</i> < <i>b</i>, <img src="Edit_15c32a7a-e9ae-41d3-8f49-c6b9c01c7ece.bmp" alt="" />(<i>t</i> small) for 2) to determine the condition for <i><i>α</i></i> and <i><i>β</i></i> to become <i>f</i>(<i>t</i>) ≤ 0 and <i>g</i>(<i>t</i>) ≥ 0.
基金Research supported by Council of Scientific and Industrial Research, India under award no.9/143(163)/91-EER-
文摘Recently Guo introduced integrated Meyer -Konig and Zeller operators and studied the rate of convergence for function of bounded variation. In this note we give a sharp estimate for these operators.
基金supported by the Faculty Research Project grant of DTU(DTU/Council/BOM-AC/Notification-/31/2018/5738)Research Fellowship from the Department of Science and Technology,New Delhi(IF170272)。
文摘We introduce a primitive class of analytic functions, by specializing in many wellknown classes, classify Ma-Minda functions based on its conditions and their interesting geometrical aspects. Further, study a newly defined subclass of starlike functions involving a special type of Ma-Minda function introduced here for obtaining inclusion and radius results. We also establish some majorization, Bloch function norms, and other related problems for the same class.
基金Grant-in-Aid for Scientific Research (C) from Japan Society for the Promotion of Science (Nos.19540205,200717540138,2007).
文摘Let B be the unit ball in a complex Banach space. Let S^＊k＋1（B） be the family of normalized starlike mappings f on B such that z = 0 is a zero of order k ＋ 1 of f（z） - z. The authors obtain sharp growth and covering theorems, as well as sharp coefficient bounds for various subsets of S^＊k＋1（B）.
基金National Natural Science Foundation of China(11971165,11561030)。
文摘Let K be the familiar class of normalized convex functions in the unit disk.In[14],Keogh and Merkes proved that for a function f(z)=z+∑k=2∞a k z k in the class K,|a 3−λa 22|≤max{13,|λ−1|},λ∈C.The above estimate is sharp for eachλ.In this article,we establish the corresponding inequality for a normalized convex function f on U such that z=0 is a zero of order k+1 of f(z)−z,and then we extend this result to higher dimensions.These results generalize some known results.
基金This work was supported by NNSF of China（Grant Nos. 11561030, 11261022）, the Jiangxi Provincial Natural Science Foundation of China （Grant Nos. 20152ACB20002, 20161BAB201019）, Natural Science Foundation of Department of Education of Jiangxi Province, China （Grant No. GJJ150301）, and the Jiangxi Provincial graduate student innovation project （Grant No. YC2016-S159）
文摘Let S~* be the familiar class of normalized univalent functions in the unit disk.In [9], Keogh and Merkes proved that for a function f(z) = z +∑k=2∞ a_kz^k in the class S~*,then |a_3-λa_2~2| ≤ max{1, |3-4λ|}, λ∈ C. In this article, we investigate the corresponding problem for the subclass of starlike mappings defined on the unit ball in a complex Banach space, on the unit polydisk in Cnand the bounded starlike circular domain in C~■, respectively.
基金supported by NSFC(Grant nos.11871406,12271449)supported by Australian Laureate Fellowship FL150100126 of the Australian Research CouncilNational Key R and D Program of China 2021YFA1001800 and NSFC(Grant no.12171334).
文摘It was conjectured by Escobar(J Funct Anal 165:101–116,1999)that for an ndimensional(n≥3)smooth compact Riemannian manifold with boundary,which has nonnegative Ricci curvature and boundary principal curvatures bounded below by c>0,the first nonzero Steklov eigenvalue is greater than or equal to c with equality holding only on isometrically Euclidean balls with radius 1/c.In this paper,we confirm this conjecture in the case of nonnegative sectional curvature.The proof is based on a combination of Qiu-Xia’s weighted Reilly-type formula with a special choice of the weight function depending on the distance function to the boundary,as well as a generalized Pohozaev-type identity.
基金supported by National Natural Science Foundation of China (Grant No.11871452)Beijing Information Science and Technology University Foundation (Grant No.2025031)+1 种基金Natural Science Foundation of Henan Province (Grant No.202300410338)the Nanhu Scholar Program for Young Scholars of Xinyang Normal University.
文摘In this paper,we calculate the sharp bound for the generalized m-linear n-dimensional Hardy-Littlewood-Polya operator on power weighted central and non-central homogeneous Morrey spaces.As an application,the sharp bound for the Hardy-Littlewood-Polya operator on power weighted central and noncentral homo-geneous Morrey spaces is obtained.Finally,we also find the sharp bound for the Hausdorff operator on power weighted central and noncentral homogeneous Morrey spaces,which generalizes the previous results.
基金supported by NSF of China(Grant Nos.10931001,10871173)supported by NSF of China(Grant No.11026104)
文摘In this paper, we study two different extensions of the Hausdorff operator to the multilinear case. Boundedness on Lebesgue spaces and Herz spaces is obtained. The bound on the Lebesgue space is optimal. Our results are substantial extensions of some known results on Multilinear high dimensional Hardy operator.
文摘We offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least n+1/n−1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size n−1/2.With the same method,we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree,provided this is at most n−1/2.
基金supported by National Natural Science Foundation of China(Grant Nos. 10931001,10901076 and 11171345)Shanghai Leading Academic Discipline Project(Grant No.J50101)supported by the Key Laboratory of Mathematics and Complex System(Beijing Normal University),Ministry of Education,China
文摘In this paper,the sharp bound for the weak-type(1,1) inequality for the n-dimensional Hardy operator is obtained.Moreover,the precise norms of generalized Hardy operators on the type of Campanato spaces are worked out.As applications,the corresponding norms of the Riemann-Liouville integral operator and the n-dimensional Hardy operator are deduced.It is also proved that the n-dimensional Hardy operator maps from the Hardy space into the Lebesgue space.The endpoint estimate for the commutator generated by the Hardy operator and the(central) BMO function is also discussed.
基金supported by National Natural Science Foundation of China(Grant Nos.11561030,11261022 and 11471111)the Jiangxi Provincial Natural Science Foundation of China(Grant Nos.20152ACB20002 and 20161BAB201019)Natural Science Foundation of Department of Education of Jiangxi Province of China(Grant No.GJJ150301)
文摘In this paper, we establish the Fekete and Szego inequality for a class of holomorphic functions in the unit disk, and then we extend this result to a class of holomorphic mappings on the unit ball in a complex Banach space or on the unit polydisk in Cn.
基金Supported by NNSF of China(Grant Nos.11561030,11471111 and 11261022)the Jiangxi Provincial Natural Science Foundation of China(Grant Nos.20152ACB20002 and 20161BAB201019)Natural Science Foundation of Department of Education of Jiangxi Province,China(Grant No.GJJ150301)
文摘Let Sα*be the familiar class of normalized starlike functions of order α in the unit disk. In this paper, we establish the Fekete and Szeg? inequality for the class Sα*, and then we generalize this result to the unit ball in a complex Banach space or on the unit polydisk in Cn.
