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We construct non-commutative analogs of transport maps among free Gibbs state satisfying a certain convexity condition. Unlike previous constructions, our approach is non-perturbative in nature and thus can be used to construct transport maps between free Gibbs states associated to potentials which are far from quadratic, i.e., states which are far from the semicircle law. An essential technical ingredient in our approach is the extension of free stochastic analysis to non-commutative spaces of functions based on the Haagerup tensor product.

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A Schwarz lemma for hyperbolic harmonic mappings in the unit ball

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-128528 ◽

2021 ◽

Vol 127 (3) ◽

Author(s):

Jiaolong Chen ◽

David Kalaj

Keyword(s):

Unit Ball ◽

Explicit Form ◽

Smooth Function ◽

Harmonic Mapping ◽

Sharp Inequality ◽

Schwarz Lemma ◽

Harmonic Mappings ◽

Manuscripta Math ◽

Sharp Constant ◽

Mapping Theory

Assume that $p\in [1,\infty ]$ and $u=P_{h}[\phi ]$, where $\phi \in L^{p}(\mathbb{S}^{n-1},\mathbb{R}^n)$ and $u(0) = 0$. Then we obtain the sharp inequality $\lvert u(x) \rvert \le G_p(\lvert x \rvert )\lVert \phi \rVert_{L^{p}}$ for some smooth function $G_p$ vanishing at $0$. Moreover, we obtain an explicit form of the sharp constant $C_p$ in the inequality $\lVert Du(0)\rVert \le C_p\lVert \phi \rVert \le C_p\lVert \phi \rVert_{L^{p}}$. These two results generalize and extend some known results from the harmonic mapping theory (D. Kalaj, Complex Anal. Oper. Theory 12 (2018), 545–554, Theorem 2.1) and the hyperbolic harmonic theory (B. Burgeth, Manuscripta Math. 77 (1992), 283–291, Theorem 1).

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Examples and classification of Riemannian submersions satisfying a basic equality

Bulletin of the Australian Mathematical Society ◽

10.1017/s000497270003522x ◽

2005 ◽

Vol 72 (3) ◽

pp. 391-402 ◽

Cited By ~ 2

Author(s):

Bang-Yen Chen

Keyword(s):

Riemannian Manifold ◽

Unit Sphere ◽

Isometric Immersion ◽

Riemannian Submersion ◽

Sharp Inequality ◽

Riemannian Submersions ◽

Equality Case ◽

Totally Geodesic ◽

Basic Equality

In an earlier article we obtain a sharp inequality for an arbitrary isometric immersion from a Riemannian manifold admitting a Riemannian submersion with totally geodesic fibres into a unit sphere. In this article we investigate the immersions which satisfy the equality case of the inequality. As a by-product, we discover a new characterisation of Cartan hypersurface in S4.

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THE SHARP BOUND FOR THE HANKEL DETERMINANT OF THE THIRD KIND FOR CONVEX FUNCTIONS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972717001125 ◽

2018 ◽

Vol 97 (3) ◽

pp. 435-445 ◽

Cited By ~ 15

Author(s):

BOGUMIŁA KOWALCZYK ◽

ADAM LECKO ◽

YOUNG JAE SIM

Keyword(s):

Convex Functions ◽

Analytic Functions ◽

Sharp Inequality ◽

Sharp Bound ◽

Hankel Determinant ◽

The Third ◽

Image Position

We prove the sharp inequality $|H_{3,1}(f)|\leq 4/135$ for convex functions, that is, for analytic functions $f$ with $a_{n}:=f^{(n)}(0)/n!,~n\in \mathbb{N}$, such that $$\begin{eqnarray}Re\bigg\{1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\bigg\}>0\quad \text{for}~z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\},\end{eqnarray}$$ where $H_{3,1}(f)$ is the third Hankel determinant $$\begin{eqnarray}H_{3,1}(f):=\left|\begin{array}{@{}ccc@{}}a_{1} & a_{2} & a_{3}\\ a_{2} & a_{3} & a_{4}\\ a_{3} & a_{4} & a_{5}\end{array}\right|.\end{eqnarray}$$

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