Some Zygmund type inequalities for the <Emphasis Type="Italic">s</Emphasis>th derivative of polynomials

For a polynomial P(z) of degree n having no zeros in |z| < 1, it was recently proved in [9] that

$$\left| {{z^s}{P^{\left( s \right)}}\left( z \right) + \beta \frac{{n\left( {n - 1} \right)...\left( {n - s + 1} \right)}}{{{2^s}}}P\left( z \right)} \right| \leqslant \frac{{n\left( {n - 1} \right)...\left( {n - s + 1} \right)}}{2}\left( {\left| {1 + \frac{\beta }{{{2^s}}}} \right| + \left| {\frac{\beta }{{{2^s}}}} \right|} \right)\mathop {\max }\limits_{\left| z \right| = 1} \left| {P\left( z \right)} \right|$$

for every β ∈ C with |β| ≤ 1, 1 ≤ s ≤ n and |z| = 1. In this paper, we obtain the L _p mean extension of the above and other related results for the sth derivative of polynomials.

     

On a Congruence Involving Generalized Fibonomial Coefficients

Let (F_n)_n≥0 be the Fibonacci sequence. For 1 ≤ k ≤ m, the Fibonomial coefficient is defined as

$${\left[ {\begin{array}{*{20}{c}} n \\ k \end{array}} \right]_F} = \frac{{{F_{n - k + 1}} \cdots {F_{n - 1}}{F_n}}}{{{F_1} \cdots {F_k}}}$$

. In 2013, Marques, Sellers and Trojovský proved that if p is a prime number such that p ≡ ±1 (mod 5), then p?\({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all integers a ≥ 1. In 2010, in particular, Kilic generalized the Fibonomial coefficients for

$${\left[ {\begin{array}{*{20}{c}} n \\ k \end{array}} \right]_{F,m}} = \frac{{{F_{\left( {n - k + 1} \right)m}} \cdots {F_{\left( {n - 1} \right)m}}{F_{nm}}}}{{{F_m} \cdots {F_{km}}}}$$

. In this note, we generalize Marques, Sellers and Trojovský result to prove, in particular, that if p ≡ ±1 (mod 5), then \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_{F,m}} \equiv 1\) (mod p), for all a ≥ 0 and m ≥ 1.

     

The <Emphasis Type="Italic">p</Emphasis>-adic order of some fibonomial coefficients whose entries are powers of <Emphasis Type="Italic">p</Emphasis>

Let (F _n ) _n≥0 be the Fibonacci sequence. For 1 ≤ k ≤ m, the Fibonomial coefficient is defined as

$${\left[ {\begin{array}{*{20}{c}} m \\ k \end{array}} \right]_F} = \frac{{{F_{m - k + 1}} \cdots {F_{m - 1}}{F_m}}}{{{F_1} \cdots {F_k}}}$$

. In 2013, Marques, Sellers and Trojovský proved that if p is a prime number such that p ≡ ±2 (mod 5), then \(p{\left| {\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]} \right._F}\) for all integers a ≥ 1. In 2015, Marques and Trojovský worked on the p-adic order of \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all a ≥ 1 when p ≠ 5. In this paper, we shall provide the exact p-adic order of \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all integers a, b ≥ 1 and for all prime number p.

     

On Schwarzian Triangle Functions,Automorphic Forms and a Generalization of Ramanujan’s Triple Differential Equations

Let G be the group of the fractional linear transformations generated by

$$T(\tau ) = \tau + \lambda ,S(\tau ) = \frac{{\tau \cos \frac{\pi }{n} + \sin \frac{\pi }{n}}}{{ - \tau \sin \frac{\pi }{n} + \cos \frac{\pi }{n}}};$$

where

$$\lambda = 2\frac{{\cos \frac{\pi }{m} + \cos \frac{\pi }{n}}}{{\sin \frac{\pi }{n}}};$$

m, n is a pair of integers with either n ≥ 2,m ≥ 3 or n ≥ 3,m ≥ 2; τ lies in the upper half plane H.

A fundamental set of functions f₀, f_i and f_∞ automorphic with respect to G will be constructed from the conformal mapping of the fundamental domain of G. We derive an analogue of Ramanujan’s triple differential equations associated with the group G and establish the connection of f₀, f_i and f_∞ with a family of hypergeometric functions.     

Estimates of sums of integrals of the Legendre Polynomial

Estimates of sums \({R_{nk}}\left( x \right) = \sum\limits_{m = n}^\infty {{P_{mk}}\left( x \right)} \) are established. Here, P_n0(x)= P_n(x), \({R_{nk}}\left( x \right) = \int\limits_.^x {{P_{n,k - 1}}\left( y \right)dy} \), P_n is the Legendre polynomial with standard normalization P_n(1) = 1. With k = 1 in the main interval [–1, 1] the sum decreases with increasing n as n^–1, and in the half-open interval [–1, 1), as n^–3/2. With k > 1 the point x = 1 does not need to be excluded. The sum decreases as n^-k–1/2. Moreover, a small increase in the multiplicative constant permits to obtain the estimate \(|{R_{nk}}\left( {\cos \theta } \right)| < \frac{{C{{\sin }^{k - 3/2}}\theta }}{{{n^{k + 1/2}}}}\), where C depends weakly on k (but not on n, θ). In passing, a Mehler–Dirichlet-type integral for R_nk(cos θ) is deduced.     

Distance-Regular Shilla Graphs with <Emphasis Type="Italic">b</Emphasis><Subscript>2</Subscript> = <Emphasis Type="Italic">c</Emphasis><Subscript>2</Subscript>

A Shilla graph is defined as a distance-regular graph of diameter 3 with second eigen-value θ₁ equal to a₃. For a Shilla graph, let us put a = a₃ and b = k/a. It is proved in this paper that a Shilla graph with b₂ = c₂ and noninteger eigenvalues has the following intersection array:

$$\left\{ {\frac{{{b^2}\left( {b - 1} \right)}}{2},\frac{{\left( {b - 1} \right)\left( {{b^2} - b + 2} \right)}}{2},\frac{{b\left( {b - 1} \right)}}{4};1,\frac{{b\left( {b - 1} \right)}}{4},\frac{{b{{\left( {b - 1} \right)}^2}}}{2}} \right\}$$

If Γ is a Q-polynomial Shilla graph with b₂ = c₂ and b = 2r, then the graph Γ has intersection array

$$\left\{ {2tr\left( {2r + 1} \right),\left( {2r + 1} \right)\left( {2rt + t + 1} \right),r\left( {r + t} \right);1,r\left( {r + t} \right),t\left( {4{r^2} - 1} \right)} \right\}$$

and, for any vertex u in Γ, the subgraph Γ₃(u) is an antipodal distance-regular graph with intersection array

$$\left\{ {t\left( {2r + 1} \right),\left( {2r - 1} \right)\left( {t + 1} \right),1;1,t + 1,t\left( {2r + 1} \right)} \right\}$$

The Shilla graphs with b₂ = c₂ and b = 4 are also classified in the paper.

     

A logarithmically improved regularity criterion for the supercritical quasi-geostrophic equations in Besov space

In this paper, we consider the logarithmically improved regularity criterion for the supercritical quasi-geostrophic equation in Besov space \(\dot B_{\infty ,\infty }^{ - r}\left( {{\mathbb{R}^2}} \right)\). The result shows that if θ is a weak solutions satisfies

$$\int_0^T {\frac{{\left\| {\nabla \theta ( \cdot ,s)} \right\|_{\dot B_{\infty ,\infty }^{ - r} }^{\tfrac{\alpha }{{\alpha - r}}} }}{{1 + \ln \left( {e + \left\| {\nabla ^ \bot \theta ( \cdot ,s)} \right\|_{L^{\tfrac{2}{r}} } } \right)!}}ds < \infty for some 0 < r < \alpha and 0 < \alpha < 1,}$$

then θ is regular at t = T. In view of the embedding \({L^{\frac{2}{r}}} \subset M_{\frac{2}{r}}^p \subset \dot B_{\infty ,\infty }^{ - r}\) with \(2 \leqslant p < \frac{2}{r}\) and 0 ≤ r < 1, we see that our result extends the results due to [20] and [31].

     

Families of Subsets Without a Given Poset in Double Chains and Boolean Lattices

Given a finite poset P, the intensively studied quantity La(n, P) denotes the largest size of a family of subsets of [n] not containing P as a weak subposet. Burcsi and Nagy (J. Graph Theory Appl. 1, 40–49 2013) proposed a double-chain method to get an upper bound \({\mathrm La}(n,P)\le \frac {1}{2}(|P|+h-2)\left (\begin {array}{c}n \\ \lfloor {n/2}\rfloor \end {array}\right )\) for any finite poset P of height h. This paper elaborates their double-chain method to obtain a new upper bound

$${\mathrm La}(n,P)\le \left( \frac{|P|+h-\alpha(G_{P})-2}{2}\right)\left( \begin{array}{c}n \\ \lfloor{\frac{n}{2}}\rfloor\end{array}\right) $$

for any graded poset P, where α(G _P ) denotes the independence number of an auxiliary graph defined in terms of P. This result enables us to find more posets which verify an important conjecture by Griggs and Lu (J. Comb. Theory (Ser. A) 119, 310–322, 2012).

     

Carleson measures,BMO spaces and balayages associated to Schrdinger operators

Let L be a Schrdinger operator of the form L =-? + V acting on L~2(R~n), n≥3, where the nonnegative potential V belongs to the reverse Hlder class B_q for some q≥n. Let BMO_L(R~n) denote the BMO space associated to the Schrdinger operator L on R~n. In this article, we show that for every f ∈ BMO_L(R~n) with compact support, then there exist g ∈ L~∞(R~n) and a finite Carleson measure μ such that f(x) = g(x) + S_(μ,P)(x) with ∥g∥∞ + |||μ|||c≤ C∥f∥BMO_L(R~n), where S_(μ,P)=∫(R_+~(n+1))Pt(x,y)dμ(y, t),and Pt(x, y) is the kernel of the Poisson semigroup {e-~(t(L)~(1/2))}t0 on L~2(R~n). Conversely, if μ is a Carleson measure, then S_(μ,P) belongs to the space BMO_L(R~n). This extends the result for the classical John-Nirenberg BMO space by Carleson(1976)(see also Garnett and Jones(1982), Uchiyama(1980) and Wilson(1988)) to the BMO setting associated to Schrdinger operators.     

Simple proofs and extensions of a result of L. D. Pustylnikov on the nonautonomous Siegel theorem

We present simple proofs of a result of L.D. Pustylnikov extending to nonautonomous dynamics the Siegel theorem of linearization of analytic mappings. We show that if a sequence f _n of analytic mappings of C ^d has a common fixed point f _n (0) = 0, and the maps f _n converge to a linear mapping A∞ so fast that

$$\sum\limits_n {{{\left\| {{f_m} - {A_\infty }} \right\|}_{L\infty \left( B \right)}} < \infty } $$

$${A_\infty } = diag\left( {{e^{2\pi i{\omega _1}}},...,{e^{2\pi i{\omega _d}}}} \right)\omega = \left( {{\omega _1},...,{\omega _q}} \right) \in {\mathbb{R}^d},$$

then f _n is nonautonomously conjugate to the linearization. That is, there exists a sequence h _n of analytic mappings fixing the origin satisfying

$${h_{n + 1}} \circ {f_n} = {A_\infty }{h_n}.$$

The key point of the result is that the functions hn are defined in a large domain and they are bounded. We show that

$${\sum\nolimits_n {\left\| {{h_n} - Id} \right\|} _{L\infty (B)}} < \infty .$$

We also provide results when f _n converges to a nonlinearizable mapping f∞ or to a nonelliptic linear mapping. In the case that the mappings f _n preserve a geometric structure (e. g., symplectic, volume, contact, Poisson, etc.), we show that the hn can be chosen so that they preserve the same geometric structure as the f _n . We present five elementary proofs based on different methods and compare them. Notably, we consider the results in the light of scattering theory. We hope that including different methods can serve as an introduction to methods to study conjugacy equations.

     

A note on some results of Hua in short intervals

In this paper, we improve the previous results of the authors [G. Lü and H. Tang, On some results of Hua in short intervals, Lith. Math. J., 50(1):54–70, 2010] by proving that each sufficiently large integer N satisfying some congruence conditions can be written as

$ \left\{ {\begin{array}{*{20}{c}} {N = p_1^2 + p_2^2 + p_3^2 + p_4^2 + {p^k},} \hfill \\ {\left| {{p_j} - \sqrt {{\frac{N}{5}}} } \right| \leqslant U,\quad \left| {p - {{\left( {\frac{N}{5}} \right)}^{\frac{1}{k}}}} \right| \leqslant U\,{N^{ - \frac{1}{2} + \frac{1}{k}}},\quad j = 1,\,2,\,\,3,\,4,} \hfill \\ \end{array} } \right. $

where U = N ^1/2?η+ε with \( \eta = \frac{1}{{2k\left( {{K^2} + 1} \right)}} \) and K = 2k ^?1, k ? 2.

     

Numerical Radius Inequalities for Certain 2 × 2 Operator Matrices

We prove several numerical radius inequalities for certain 2 × 2 operator matrices. Among other inequalities, it is shown that if X, Y, Z, and W are bounded linear operators on a Hilbert space, then

$$w\left( \left[\begin{array}{cc} X &; Y \\ Z &; W \end{array} \right] \right) \geq \max \left(w(X),w(W),\frac{w(Y+Z)}{2},\frac{w(Y-Z)}{2}\right) $$

and

$$w\left( \left[\begin{array}{cc}X &; Y \\ Z &; W\end{array} \right] \right) \leq \max \left( w(X), w(W)\right)+\frac{w(Y+Z)+w(Y-Z)}{2}. $$

As an application of a special case of the second inequality, it is shown that

$$\frac{\left\Vert X\right\Vert }{2}+\frac{\left\vert \left\Vert\operatorname{Re}{X}\right\Vert -\frac{\left\Vert X\right\Vert}{2}\right\vert }{4}+\frac{ \left\vert \left\Vert \operatorname{Im}{X}\right\Vert -\frac{\left\Vert X\right\Vert}{2}\right\vert }{4} \leq w(X), $$

which is a considerable improvement of the classical inequality \({\frac{ \left\Vert X\right\Vert }{2}\leq w(X)}\) . Here w(·) and || · || are the numerical radius and the usual operator norm, respectively.

     

A remark on the existence of entire large and bounded solutions to a (<Emphasis Type="Italic">k</Emphasis><Subscript>1</Subscript>, <Emphasis Type="Italic">k</Emphasis><Subscript>2</Subscript>)-Hessian system with gradient term

In this paper, we study the existence of positive entire large and bounded radial positive solutions for the following nonlinear system

$$\left\{ {\begin{array}{*{20}c}{S_{k_1 } \left( {\lambda \left( {D^2 u_1 } \right)} \right) + a_1 \left( {\left| x \right|} \right)\left| {\nabla u_1 } \right|^{k_1 } = p_1 \left( {\left| x \right|} \right)f_1 \left( {u_2 } \right)} & {for x \in \mathbb{R}^N ,} \\{S_{k_2 } \left( {\lambda \left( {D^2 u_2 } \right)} \right) + a_2 \left( {\left| x \right|} \right)\left| {\nabla u_2 } \right|^{k_2 } = p_2 \left( {\left| x \right|} \right)f_2 \left( {u_1 } \right)} & {for x \in \mathbb{R}^N .} \\\end{array} } \right.$$

Here \({S_{{k_i}}}\left( {\lambda \left( {{D^2}{u_i}} \right)} \right)\) is the k _i -Hessian operator, a ₁, p ₁, f ₁, a ₂, p ₂ and f ₂ are continuous functions.

     

<Emphasis Type="Italic">L</Emphasis>-Approximation of a linear combination of the poisson kernel and its conjugate kernel by trigonometric polynomials

A linear combination Π _q,α = cos(απ/2)P + sin(απ/2)Q of the Poisson kernel P(t) = 1/2 + q cos t + q ² cos 2t + ... and its conjugate kernel Q(t) = q sin t + q ² sin 2t + ... is considered for α ∈ ? and |q| < 1. A new explicit formula is found for the value E _n?1(Π _q,α ) of the best approximation in the space L = L _2π of the function Π _q,α by the subspace of trigonometric polynomials of order at most n ? 1. More exactly, it is proved that \(E_{n - 1} \left( {\prod _{q,\alpha } } \right) = \left. {\frac{{\left| q \right|^n \left( {1 - q^2 } \right)}}{{1 - q^{4n} }}} \right\|\left. {\frac{{\cos \left( {nt - {{\alpha \pi } \mathord{\left/ {\vphantom {{\alpha \pi } 2}} \right. \kern-\nulldelimiterspace} 2}} \right) - q^{2n} \cos \left( {nt + {{\alpha \pi } \mathord{\left/ {\vphantom {{\alpha \pi } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}}{{1 - q^2 - 2q \cos t}}} \right\|_L\). In addition, the value E _n?1(Π _q,α ) is represented as a rapidly convergent series.     

Weighted polyharmonic equation with Navier boundary conditions in a half space

We study positive solutions of the following polyharmonic equation with Hardy weights associated to Navier boundary conditions on a half space:?????(-?)~mu(x)=u~p(x)/|x|~s,in R_+~n,u(x)=-?u(x)=…=(-?)~(m-1)u(x)=0,on ?R_+~n,(0.1)where m is any positive integer satisfying 02mn.We first prove that the positive solutions of(0.1)are super polyharmonic,i.e.,(-?)~iu0,i=0,1,...,m-1.(0.2) For α=2m,applying this important property,we establish the equivalence between (0.1) and the integral equation u(x)=c_n∫R_+~n(1/|x-y|~(n-α)-1/|x~*-y|~(n-α))u~p(y)/|y|~sdy,(0.3) where x~*=(x1,...,x_(n-1),-x_n) is the reflection of the point x about the plane R~(n-1).Then,we use the method of moving planes in integral forms to derive rotational symmetry and monotonicity for the positive solution of(0.3),in whichαcan be any real number between 0 and n.By some Pohozaev type identities in integral forms,we prove a Liouville type theorem—the non-existence of positive solutions for(0.1).     

Strong isochronism of a center and a focus for systems with homogeneous nonlinearities

We suggest a new approach to studying the isochronism of the system

${{dx} \mathord{\left/ {\vphantom {{dx} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = - y + p_n (x,y),{{dy} \mathord{\left/ {\vphantom {{dy} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = x + q_n (x,y),$

where p _n and q _n are homogeneous polynomials of degree n. This approach is based on the normal form

${{dX} \mathord{\left/ {\vphantom {{dX} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = - Y + XS(X,Y),{{dY} \mathord{\left/ {\vphantom {{dY} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = X + YS(X,Y)$

and its analog in polar coordinates. We prove a theorem on sufficient conditions for the strong isochronism of a center and a focus for the reduced system and obtain examples of centers with strong isochronism of degrees n = 4, 5. The present paper is the first to give examples of foci with strong isochronism for the system in question.

     

The hole probability for Gaussian entire functions

Consider the random entire function

$f(z) = \sum\limits_{n = 0}^\infty {{\phi _n}{a_n}{z^n}} $

, where the ? _n are independent standard complex Gaussian coefficients, and the a _n are positive constants, which satisfy

$\mathop {\lim }\limits_{x \to \infty } {{\log {a_n}} \over n} = - \infty $

.

We study the probability P _H (r) that f has no zeroes in the disk{|z| < r} (hole probability). Assuming that the sequence a _n is logarithmically concave, we prove that

$\log {P_H}(r) = - S(r) + o(S(r))$

, where

$S(r) = 2 \cdot \sum\limits_{n:{a_n}{r^n} \ge 1} {\log ({a_n}{r^n})} $

, and r tends to ∞ outside a (deterministic) exceptional set of finite logarithmic measure.

     

Packing spanning graphs from separable families

Let \(\mathcal{G}\) be a separable family of graphs. Then for all positive constants ? and Δ and for every sufficiently large integer n, every sequence G ₁,..., G _t ∈ \(\mathcal{G}\) of graphs of order n and maximum degree at most Δ such that

$$\left( {{G_1}} \right) + \cdots + e\left( {{G_t}} \right) \leqslant \left( {1 - \epsilon } \right)\left( {\begin{array}{*{20}{c}}n \\ 2 \end{array}} \right)$$

packs into K _n . This improves results of Böttcher, Hladký, Piguet and Taraz when \(\mathcal{G}\) is the class of trees and of Messuti, Rödl, and Schacht in the case of a general separable family. The result also implies approximate versions of the Oberwolfach problem and of the Tree Packing Conjecture of Gyárfás and Lehel (1976) for the case that all trees have maximum degree at most Δ.

The proof uses the local resilience of random graphs and a special multi-stage packing procedure.     

Two-weight inequality for operator-valued positive dyadic operators by parallel stopping cubes

We study the operator-valued positive dyadic operator

$${T_\lambda }\left( {f\sigma } \right): = \sum\limits_{Q \in D} {{\lambda _Q}} \int_Q {fd\sigma 1Q}, $$

where the coefficients {λ _Q : C → D} _Q∈D are positive operators from a Banach lattice C to a Banach lattice D. We assume that the Banach lattices C and D* each have the Hardy–Littlewood property. An example of a Banach lattice with the Hardy–Littlewood property is a Lebesgue space.

In the two-weight case, we prove that the L _C ^p (σ) → L _D ^q (ω) boundedness of the operator T _λ( · σ) is characterized by the direct and the dual L ^∞ testing conditions:

$$\left\| {{1_Q}{T_\lambda }} \right\|{\left( {{1_Q}f\sigma } \right)||_{L_D^q\left( \omega \right)}} \lesssim {\left\| f \right\|_{L_C^\infty \left( {Q,\sigma } \right)}}\sigma {\left( Q \right)^{1/p}}$$

,

$${\left\| {{1_Q}{T_\lambda }*\left( {{1_{Qg\omega }}} \right)} \right\|_{L_{C*}^{p'}\left( \sigma \right)}} \lesssim {\left\| g \right\|_{L_{D*}^\infty \left( {Q,\omega } \right)}}\omega {\left( Q \right)^{1/q'}}$$

.

Here L _C ^p (σ) and L _D ^q (ω) denote the Lebesgue–Bochner spaces associated with exponents 1 < p ≤ q < ∞, and locally finite Borel measures σ and ω.

In the unweighted case, we show that the L _C ^p (μ) → L _D ^p (μ) boundedness of the operator T _λ( · μ) is equivalent to the end-point direct L ^∞ testing condition:

$${\left\| {{1_Q}{T_\lambda }\left( {{1_Q}f\mu } \right)} \right\|_{L_D^1\left( \mu \right)}} \lesssim {\left\| f \right\|_{L_C^\infty \left( {Q,\mu } \right)}}\left( {Q,\mu } \right)\mu \left( Q \right)$$

.

This condition is manifestly independent of the exponent p. By specializing this to particular cases, we recover some earlier results in a unified way.     

An Improvement of the General Bound on the Largest Family of Subsets Avoiding a Subposet

Let La(n, P) be the maximum size of a family of subsets of [n] = {1, 2, … , n} not containing P as a (weak) subposet, and let h(P) be the length of a longest chain in P. The best known upper bound for La(n, P) in terms of |P| and h(P) is due to Chen and Li, who showed that \(\text {La}(n,P) \le \frac {1}{m+1} \left (|{P}| + \frac {1}{2}(m^{2} +3m-2)(h(P)-1) -1 \right ) {\left (\begin {array}{c}{n}\\ {\lfloor n/2 \rfloor } \end {array}\right )}\) for any fixed m ≥ 1. In this paper we show that \(\text {La}(n,P) \le \frac {1}{2^{k-1}} \left (|P| + (3k-5)2^{k-2}(h(P)-1) - 1 \right ) {\left (\begin {array}{c}{n}\\ {\lfloor n/2 \rfloor } \end {array}\right )}\) for any fixed k ≥ 2, improving the best known upper bound. By choosing k appropriately, we obtain that \(\text {La}(n,P) = \mathcal {O}\left (h(P) \log _{2}\left (\frac {|{P}|}{h(P)}+2\right ) \right ) {\left (\begin {array}{c}{n}\\ {\lfloor n/2 \rfloor } \end {array}\right )}\) as a corollary, which we show is best possible for general P. We also give a different proof of this corollary by using bounds for generalized diamonds. We also show that the Lubell function of a family of subsets of [n] not containing P as an induced subposet is \(\mathcal {O}(n^{c})\) for every \(c>\frac {1}{2}\).