Blow Up Criteria for the Incompressible Nematic Liquid Crystal Flows

In this paper, we investigate blow up criteria for the local smooth solutions to the 3D incompressible nematic liquid crystal flows via the components of the gradient velocity field \(\nabla u\) and the gradient orientation field \(\nabla d\). More precisely, we show that \(0< T_{ \ast}<+\infty\) is the maximal time interval if and only if

$$\begin{aligned} & \int_{0}^{T_{\ast}} \bigl\Vert \Vert \partial_{i}u\Vert _{L_{x_{i}} ^{\gamma}} \bigr\Vert _{L_{x_{j}x_{k}}^{\alpha}}^{\beta}+ \|\nabla d\| _{L^{\infty}}^{\frac{8}{3}}\mathrm{d}t=\infty, \\ &\quad\text{ with } \frac{2}{\alpha}+\frac{2}{\beta}\leq\frac{3\alpha +2}{4\alpha}, \text{ and } 1\leq\gamma\leq\alpha,2< \alpha\leq+\infty, \end{aligned}$$

or

$$\begin{aligned} \int_{0}^{T_{\ast}}\|\partial_{3}u_{3} \|^{\beta}_{L^{\alpha}}+\| \nabla d\|^{\frac{8}{3}}_{L^{\infty}} \mathrm{d}t=\infty,\quad\text{with } \frac{3}{\alpha}+\frac{2}{\beta}\leq \frac{3(\alpha+2)}{4 \alpha}, \text{ and } 2< \alpha\leq\infty, \end{aligned}$$

where \(i,j,k\in\{1,2,3\}\), \(i\neq j\), \(i\neq k\), and \(j\neq k\).

     

Mean Square Estimates for Coefficients of Symmetric Power <Emphasis Type="Italic">L</Emphasis>-Functions

Let L(sym ^j f,s) be the jth symmetric power L-function attached to a holomorphic Hecke eigencuspform f(z) for the full modular group, and \(\lambda_{\mathrm{sym}^{j}f}(n)\) denote its nth coefficient. In this paper we are able to prove that

$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{3}f}(n)\bigg|^{2}dy=O\bigl(x^{2}\bigr),$

and

$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{4}f}(n)\bigg|^{2}dy=O\bigl(x^{\frac{11}{5}}\log x\bigr).$

     

Nearly Quartic Mappings in β-Homogeneous F-Spaces

In this paper, we investigate the Hyers–Ulam stability of the following quartic equation $$\begin{array}{ll} {\sum\limits^{n}_{k=2}}\left({\sum\limits^{k}_{i_{1}=2}}{\sum\limits^{k+1}_{i_{2}=i_{1}+1}} \ldots {\sum\limits^{n}_{i_{n-k+1}=i_{n-k}+1}}\right)\\ \quad\times f \left({\sum\limits^{n}_{i=1,i \neq i_{1},\ldots,i_{n-k+1}}} x_{i}-{\sum\limits^{n-k+1}_{r=1}}x_{i_{r}}\right) + f \left({\sum\limits^{n}_{i=1}}x_{i}\right)\\ \quad-2^{n-2}{\sum\limits^{}_{1 \leq{i} \leq{j} \leq{n}}}(f(x_{i} + x_{j}){+f(x_{i} - x_{j})){+2^{n-5}(n - 2){\sum\limits^{n}_{i=1}}f(2x_{i})}} = \theta \end{array} $$ $({n \in \mathbb{N}, n \geq 3})$ in β-homogeneous F-spaces.     

Polynomiality of some hook-length statistics

We prove a conjecture of Okada giving an exact formula for a certain statistic for hook-lengths of partitions:

$\frac{1}{n!} \sum_{\lambda \vdash n} f_{\lambda}^2 \sum_{u \in \lambda} \prod_{i=1}^{r}\bigl(h_u^2 - i^2\bigr) = \frac{1}{2(r+1)^2} \binom{2r}{r}\binom{2r+2}{ r+1} \prod_{j=0}^{r} (n-j),$

where f _λ is the number of standard Young tableaux of shape λ and h _u is the hook length of the square u of the Young diagram of λ. We also obtain other similar formulas.

     

On the stability of a generalization of Jensen functional equation

Using the fixed point method, we investigate the stability of a generalization of Jensen functional equation

$$ \sum_{k=0}^{n-1} f(x+ b_{k}y)=nf(x),$$

where \({n \in \mathbb{N}_{2}}\), \({b_{k}=\exp(\frac{2i\pi k}{n})}\) for \({0\leq k \leq n-1}\), in Banach spaces. Also, we prove the hyperstability results of this equation by the fixed point method.

     

On the optimal interpoint distance sum inequality

Let \({{\left\{x_{1}, \dots, x_{n}\right\}\subset \mathbb{R}^2}}\) be a set of points in the unit circle. It is shown that

$\sum\limits^{n}_{i=1}{\min_{j \neq i}{\left\|x_{i} - x_{j}\right\|^2}}\leq9,$

which is best possible and improves earlier results by Arpacioglu and Haas and Xia and Liu.

     

A Refinement of the Kolmogorov–Marcinkiewicz–Zygmund Strong Law of Large Numbers

Let {X _n ; n≥1} be a sequence of independent copies of a real-valued random variable X and set S _n =X ₁+???+X _n , n≥1. This paper is devoted to a refinement of the classical Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers. We show that for 0<p<2,

$\sum_{n=1}^{\infty}\frac{1}{n}\biggl(\frac{|S_{n}|}{n^{1/p}}\biggr)<\infty\quad \mbox{almost surely}$

if and only if

$\begin{cases}\mathbb{E}|X|^{p}<\infty &; \mbox{if }0 < p < 1,\\\mathbb{E}X=0,\ \sum_{n=1}^{\infty}\frac{|\mathbb{E}XI\{|X|\leq n\}|}{n}<\infty,\mbox{ and }\\\sum_{n=1}^{\infty}\frac{\int_{\min\{u_{n},n\}}^{n}\mathbb{P}(|X|>t)\,dt}{n}<\infty &; \mbox{if }p = 1,\\\mathbb{E}X=0\mbox{ and }\int_{0}^{\infty}\mathbb{P}^{1/p}(|X|>t)\,dt<\infty,&;\mbox{if }1 < p < 2,\end{cases}$

where \(u_{n}=\inf \{t:~\mathbb{P}(|X|>t)<\frac{1}{n}\}\), n≥1. Versions of the above result in a Banach space setting are also presented. To establish these results, we invoke the remarkable Hoffmann-Jørgensen (Stud. Math. 52:159–186, 1974) inequality to obtain some general results for sums of the form \(\sum_{n=1}^{\infty}a_{n}\|\sum_{i=1}^{n}V_{i}\|\) (where {V _n ; n≥1} is a sequence of independent Banach-space-valued random variables, and a _n ≥0, n≥1), which may be of independent interest, but which we apply to \(\sum_{n=1}^{\infty}\frac{1}{n}(\frac{|S_{n}|}{n^{1/p}})\).

     

A Geometric Uncertainty Principle with an Application to Pleijel’s Estimate

Let \({\Omega \subset \mathbb{R}^2}\) be an open, bounded domain and \({\Omega = \bigcup_{i = 1}^{N} \Omega_{i}}\) be a partition. Denote the Fraenkel asymmetry by \({0 \leq \mathcal{A}(\Omega_i) \leq 2}\) and write $$D(\Omega_i) := \frac{|\Omega_{i}| - {\rm min}_{1 \leq j \leq N}{|\Omega_{j}|}}{|\Omega_{i}|}$$ with \({0 \leq D(\Omega_{i}) \leq 1}\) . For N sufficiently large depending only on \({\Omega}\) , there is an uncertainty principle $$\left(\sum_{i=1}^{N}{\frac{|\Omega_{i}|}{|\Omega|}{\mathcal{A}}(\Omega_i)}\right) + \left(\sum_{i=1}^{N}{\frac{|\Omega_i|}{|\Omega|}D(\Omega_i)}\right) \geq \frac{1}{60000}.$$ The statement remains true in dimensions \({n \geq 3}\) for some constant \({c_{n} > 0}\) . As an application, we give an (unspecified) improvement of Pleijel’s estimate on the number of nodal domains of a Laplacian eigenfunction and an improved inequality for a spectral partition problem.     

The Heat Kernel for Kolmogorov Type Operators and its Applications

There are two parts of this article. We first find explicit formulas for the heat kernel of the sub-elliptic operators $\frac{1}{2}\partial_{x}^{2}-x^{m}\partial_{y}$ with m=0,1,2. We also find the heat kernel for the sub-elliptic operator $\frac{1}{2}\sum_{j=1}^{n}\partial_{x_{j}}^{2}+(\sum_{j=1}^{n}a_{j}x_{j})\partial_{y}$ , with a _i constants. In the second part of this paper, we apply results from the first part to construct a close form formula for pricing Asian options on a geometric moving average.     

Fractional sums and Euler-like identities

We introduce a natural definition for sums of the form

$\sum_{\nu=1}^xf(\nu)$

when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the Γ function or Euler’s little-known formula \(\sum_{\nu=1}^{-1/2}\frac{1}{\nu}=-2\ln 2\).

Many classical identities like the geometric series and the binomial theorem nicely extend to this more general setting. Sums with a fractional number of terms are closely related to special functions, in particular the Riemann and Hurwitz ζ functions. A number of results about fractional sums can be interpreted as classical infinite sums or products or as limits, including identities like

$\begin{array}{l}\displaystyle\lim_{n\to\infty}\Biggl[e^{\frac{n}{4}(4n+1)}n^{-\frac{1}{8}-n(n+1)}(2\pi)^{-\frac{n}{2}}\prod_{k=1}^{2n}\Gamma\biggl(1+\frac{k}{2}\biggr)^{k(-1)^k}\Biggr]\\[12pt]\quad =\displaystyle\sqrt[12]{2}\exp\biggl(\frac{5}{24}-\frac{3}{2}\zeta'(-1)-\frac{7\zeta(3)}{16\pi^2}\biggr),\end{array}$

some of which seem to be new; and even for those which are known, our approach provides a new method to derive these identities and many others.

     

Stability of a quartic functional equation

In this paper, the authors investigate the general solution and generalized Hyers–Ulam stability of the n-dimensional quartic functional equation of the form

$$\begin{aligned} f\left( \sum _{i=1}^{n}x_i\right)&= \sum _{1 \le i<j< k< l\le n} f\left( x_i+x_j+x_k+x_l\right) +\left( -n+4\right) \nonumber \\ {}&\sum _{1 \le i< j< k \le n} f\left( x_i+x_j+x_k\right) +\left( \frac{n^2-7n+12}{2}\right) \sum _{ \begin{array}{c} 1=i;\\ i\ne j \end{array}}^{n} f\left( x_i+x_j\right) \nonumber \\&- \sum _{i=1}^{n} f\left( 2x_i\right) + \left( \frac{-n^3+9n^2-26n+120}{6}\right) \ \ \sum _{i=1}^{n}\left( \frac{f(x_i)+f(-x_i)}{2}\right) \end{aligned}$$

where n is a positive integer with \({\mathbb {N}}- \{0,1,2,3,4\}\). The stability of this quartic functional equation is introduced in Banach space using direct and fixed point methods.

     

Continued fractions for some transcendental numbers

We consider series of the form

$$\begin{aligned} \frac{p}{q} +\sum _{j=2}^\infty \frac{1}{x_j}, \end{aligned}$$

where \(x_1=q\) and the integer sequence \((x_n)\) satisfies a certain non-autonomous recurrence of second order, which entails that \(x_n|x_{n+1}\) for \(n\ge 1\). It is shown that the terms of the sequence, and multiples of the ratios of successive terms, appear interlaced in the continued fraction expansion of the sum of the series, which is a transcendental number.

     

Sharp Inequalities Involving $$(n!)^{1/n}$$

We prove that, for all integers \(n\ge 1\),

$$\begin{aligned} \Big (\sqrt{2\pi n}\Big )^{\frac{1}{n(n+1)}}\left( 1-\frac{1}{n+a}\right) <\frac{\root n \of {n!}}{\root n+1 \of {(n+1)!}}\le \Big (\sqrt{2\pi n}\Big )^{\frac{1}{n(n+1)}}\left( 1-\frac{1}{n+b}\right) \end{aligned}$$

and

$$\begin{aligned} \big (\sqrt{2\pi n}\big )^{1/n}\left( 1-\frac{1}{2n+\alpha }\right) <\left( 1+\frac{1}{n}\right) ^{n}\frac{\root n \of {n!}}{n}\le \big (\sqrt{2\pi n}\big )^{1/n}\left( 1-\frac{1}{2n+\beta }\right) , \end{aligned}$$

with the best possible constants

$$\begin{aligned}&a=\frac{1}{2},\quad b=\frac{1}{2^{3/4}\pi ^{1/4}-1}=0.807\ldots ,\quad \alpha =\frac{13}{6} \\&\text {and}\quad \beta =\frac{2\sqrt{2}-\sqrt{\pi }}{\sqrt{\pi }-\sqrt{2}}=2.947\ldots . \end{aligned}$$

     

On the entire self-shrinking solutions to Lagrangian mean curvature flow

The authors prove that the logarithmic Monge?CAmpère flow with uniformly bound and convex initial data satisfies uniform decay estimates away from time t?=?0. Then applying the decay estimates, we conclude that every entire classical strictly convex solution of the equation $$ \det D^{2}u=\exp\left\{n\left(-u+\frac{1}{2} \sum_{i=1}^{n}x_{i} \frac{\partial u}{\partial x_{i}} \right)\right\}, $$ should be a quadratic polynomial if the inferior limit of the smallest eigenvalue of the function |x|² D ² u at infinity has an uniform positive lower bound larger than 2(1 ? 1/n). Using a similar method, we can prove that every classical convex or concave solution of the equation $$ \sum_{i=1}^{n} \arctan\lambda_{i}=-u+\frac{1}{2} \sum_{i=1}^{n}x_{i} \frac{\partial u}{\partial x_{i}} $$ must be a quadratic polynomial, where ?? _i are the eigenvalues of the Hessian D ² u.     

Irrational Factor of Order k and ITS Connections With k-Free Integers

We introduce an irrational factor of order k defined by \({I_{k}(n) ={\prod_{i=1}^{l}} p_{i}^{\beta_{i}}}\) , where \({n = \prod_{i=1}^{l} p_{i}^{\alpha_{i}}}\) is the factorization of n and \({\beta_{i} = \left\{\begin{array}{ll}\alpha_i, \quad \quad {\rm if} \quad \alpha_i < k \\ \frac{1}{\alpha_i},\quad \quad {\rm if} \quad \alpha_i \geqq k \end{array}\right.}\) . It turns out that the function \({\frac{I_{k} (n)}{n}}\) well approximates the characteristic function of k-free integers. We also derive asymptotic formulas for \({\prod_{v=1}^{n} I_{k}(v)^{\frac{1}{n}}, \sum_{n \leqq x} I_{k}(n)}\) and \({\sum_{n \leqq x} (1 - \frac{n}{x}) I_{k}(n)}\) .     

Harnack Inequality for Non-Local Schrödinger Operators

Let \(x \in \mathbb {R}^{d}\), d ≥ 3, and \(f: \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a twice differentiable function with all second partial derivatives being continuous. For 1 ≤ i, j ≤ d, let \(a_{ij} : \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrödinger operator associated to

$$\mathcal{L}f(x) = \frac12 \sum\limits_{i=1}^{d} \sum\limits_{j=1}^{d} \frac{\partial}{\partial x_{i}} \left( a_{ij}(\cdot) \frac{\partial f}{\partial x_{j}}\right)(x) + {\int}_{\mathbb{R}^{d}\setminus{\{0\}}} [f(y) - f(x) ]J(x,y)dy $$

where \(J: \mathbb {R}^{d} \times \mathbb {R}^{d} \rightarrow \mathbb {R}\) is a symmetric measurable function. Let \(q: \mathbb {R}^{d} \rightarrow \mathbb {R}.\) We specify assumptions on a, q, and J so that non-negative bounded solutions to

$$\mathcal{L}f + qf = 0 $$

satisfy a Harnack inequality. As tools we also prove a Carleson estimate, a uniform Boundary Harnack Principle and a 3G inequality for solutions to \(\mathcal {L}f = 0.\)

     

Sharp Affine and Improved Moser–Trudinger–Adams Type Inequalities on Unbounded Domains in the Spirit of Lions

The purpose of this paper is threefold. First, we prove sharp singular affine Moser–Trudinger inequalities on both bounded and unbounded domains in \({\mathbb {R}}^{n}\). In particular, we will prove the following much sharper affine Moser–Trudinger inequality in the spirit of Lions (Rev Mat Iberoamericana 1(2):45–121, 1985) (see our Theorem 1.4): Let \(\alpha _{n}=n\left( \frac{n\pi ^{\frac{n}{2}}}{\Gamma (\frac{n}{2}+1)}\right) ^{\frac{1}{n-1}}\), \(0\le \beta <n\) and \(\tau >0\). Then there exists a constant \(C=C\left( n,\beta \right) >0\) such that for all \(0\le \alpha \le \left( 1-\frac{\beta }{n}\right) \alpha _{n}\) and \(u\in C_{0}^{\infty }\left( {\mathbb {R}}^{n}\right) \setminus \left\{ 0\right\} \) with the affine energy \(~{\mathcal {E}}_{n}\left( u\right) <1\), we have

$$\begin{aligned} {\displaystyle \int \nolimits _{{\mathbb {R}}^{n}}} \frac{\phi _{n,1}\left( \frac{2^{\frac{1}{n-1}}\alpha }{\left( 1+{\mathcal {E}}_{n}\left( u\right) ^{n}\right) ^{\frac{1}{n-1}}}\left| u\right| ^{\frac{n}{n-1}}\right) }{\left| x\right| ^{\beta }}dx\le C\left( n,\beta \right) \frac{\left\| u\right\| _{n}^{n-\beta }}{\left| 1-{\mathcal {E}}_{n}\left( u\right) ^{n}\right| ^{1-\frac{\beta }{n}}}. \end{aligned}$$

Moreover, the constant \(\left( 1-\frac{\beta }{n}\right) \alpha _{n}\) is the best possible in the sense that there is no uniform constant \(C(n, \beta )\) independent of u in the above inequality when \(\alpha >\left( 1-\frac{\beta }{n}\right) \alpha _{n}\). Second, we establish the following improved Adams type inequality in the spirit of Lions (Theorem 1.8): Let \(0\le \beta <2m\) and \(\tau >0\). Then there exists a constant \(C=C\left( m,\beta ,\tau \right) >0\) such that

$$\begin{aligned} \underset{u\in W^{2,m}\left( {\mathbb {R}}^{2m}\right) , \int _{ {\mathbb {R}}^{2m}}\left| \Delta u\right| ^{m}+\tau \left| u\right| ^{m} \le 1}{\sup } {\displaystyle \int \nolimits _{{\mathbb {R}}^{2m}}} \frac{\phi _{2m,2}\left( \frac{2^{\frac{1}{m-1}}\alpha }{\left( 1+\left\| \Delta u\right\| _{m}^{m}\right) ^{\frac{1}{m-1}}}\left| u\right| ^{\frac{m}{m-1}}\right) }{\left| x\right| ^{\beta }}dx\le C\left( m,\beta ,\tau \right) , \end{aligned}$$

for all \(0\le \alpha \le \left( 1-\frac{\beta }{2m}\right) \beta (2m,2)\). When \(\alpha >\left( 1-\frac{\beta }{2m}\right) \beta (2m,2)\), the supremum is infinite. In the above, we use

$$\begin{aligned} \phi _{p,q}(t)=e^{t}- {\displaystyle \sum \limits _{j=0}^{j_{\frac{p}{q}}-2}} \frac{t^{j}}{j!},\,\,\,j_{\frac{p}{q}}=\min \left\{ j\in {\mathbb {N}} :j\ge \frac{p}{q}\right\} \ge \frac{p}{q}. \end{aligned}$$

The main difficulties of proving the above results are that the symmetrization method does not work. Therefore, our main ideas are to develop a rearrangement-free argument in the spirit of Lam and Lu (J Differ Equ 255(3):298–325, 2013; Adv Math 231(6): 3259–3287, 2012), Lam et al. (Nonlinear Anal 95: 77–92, 2014) to establish such theorems. Third, as an application, we will study the existence of weak solutions to the biharmonic equation

$$\begin{aligned} \left\{ \begin{array}{l} \Delta ^{2}u+V(x)u=f(x,u)\text { in }{\mathbb {R}}^{4}\\ u\in H^{2}\left( {\mathbb {R}}^{4}\right) ,~u\ge 0 \end{array} \right. , \end{aligned}$$

where the nonlinearity f has the critical exponential growth.

     

Spectrum and the trace formula for a two-dimensional Schrödinger operator in a homogeneous magnetic field

In the space L ₂(?²), we consider the operator

$H = \left( {\frac{1}{i}\frac{\partial }{{\partial x_1 }} - x_2 } \right)^2 + \left( {\frac{1}{i}\frac{\partial }{{\partial x_2 }} + x_1 } \right)^2 + V,V = V(x) \in L_2 (\mathbb{R}^2 ).$

. We study the spectrum of H and, for V ∈ C ₀ ² (?²), prove the trace formula

$\sum\limits_{k = 0}^\infty {\left( {\sum\limits_{i = - k}^\infty {(4k + 2 - \mu _k^{(i)} ) + c_0 } } \right)} = \frac{1}{{8\pi }}\int\limits_{\mathbb{R}^2 } {V^2 (x)dx,} $

where c ₀ = π ^?1 \(\smallint _{\mathbb{R}^2 } \) V(x) dx and the µ _k ⁽ⁱ⁾ are the eigenvalues of H.

     

Maximal inequalities and Riesz transform estimates on L p spaces for magnetic Schrödinger operators II

The paper concerns the magnetic Schrödinger operator ${H({\bf a},V)=\sum_{j=1}^{n} (\frac{1}{i}\frac{\partial}{\partial x_{j}}-a_{j})^{2}+V }$ on ${\mathbb{R}^n}$ . We prove some L ^p estimates on the Riesz transforms of H and we establish some related maximal inequalities. The conditions that we arrive at, are essentially based on the control of the magnetic field by the electric potential.     

Sharp Moser–Trudinger Inequalities on Hyperbolic Spaces with Exact Growth Condition

Let \(\Phi _{n}(x)=e^x-\sum _{j=0}^{n-2}\frac{x^j}{j!}\) and \(\alpha _{n} =n\omega _{n-1}^{\frac{1}{n-1}}\) be the sharp constant in Moser’s inequality (where \(\omega _{n-1}\) is the area of the surface of the unit \(n\)-ball in \(\mathbb {R}^n\)), and \(dV\) be the volume element on the \(n\)-dimensional hyperbolic space \((\mathbb {H}^n, g)\) (\(n\ge {2}\)). In this paper, we establish the following sharp Moser–Trudinger type inequalities with the exact growth condition on \(\mathbb {H}^n\):

For any \(u\in {W^{1,n}(\mathbb {H}^n)}\) satisfying \(\Vert \nabla _{g}u\Vert _{n}\le {1}\), there exists a constant \(C(n)>0\) such that

$$\begin{aligned} \int _{\mathbb {H}^n}\frac{\Phi _{n}(\alpha _{n}|u|^{\frac{n}{n-1}})}{(1+|u|)^{\frac{n}{n-1}}}dV \le {C(n)\Vert u\Vert _{L^n}^{n}}. \end{aligned}$$

The power \(\frac{n}{n-1}\) and the constant \(\alpha _{n}\) are optimal in the following senses:

1. (i)
   If the power \(\frac{n}{n-1}\) in the denominator is replaced by any \(p<\frac{n}{n-1}\), then there exists a sequence of functions \(\{u_{k}\}\) such that \(\Vert \nabla _{g}u_{k}\Vert _{n}\le {1}\), but

   $$\begin{aligned} \frac{1}{\Vert u_{k}\Vert _{L^n}^{n}}\int _{\mathbb {H}^n} \frac{\Phi _{n}(\alpha _{n}(|u_{k}|)^{\frac{n}{n-1}})}{(1+|u_{k}|)^{p}}dV \rightarrow {\infty }. \end{aligned}$$
    
2. (ii)
   If \(\alpha >\alpha _{n}\), then there exists a sequence of function \(\{u_{k}\}\) such that \(\Vert \nabla _{g}u_{k}\Vert _{n}\le {1}\), but

   $$\begin{aligned} \frac{1}{\Vert u_{k}\Vert _{L^n}^{n}}\int _{\mathbb {H}^n} \frac{\Phi _{n}(\alpha (|u_{k}|)^{\frac{n}{n-1}})}{(1+|u_{k}|)^{p}}dV\rightarrow {\infty }, \end{aligned}$$

   for any \(p\ge {0}\).
    

This result sharpens the earlier work of the authors Lu and Tang (Adv Nonlinear Stud 13(4):1035–1052, 2013) on best constants for the Moser–Trudinger inequalities on hyperbolic spaces.