Parameter convexity and concavity of generalized trigonometric functions

We study the convexity properties of the generalized trigonometric functions viewed as functions of the parameter. We show that p→_sinp?(y)$p\to {\sin }_p?(y)$ and p→_cosp?(y)$p\to {\cos }_p?(y)$ are log-concave on the appropriate intervals while p→_tanp?(y)$p\to {\tan }_p?(y)$ is log-convex. We also prove similar facts about the generalized hyperbolic functions. In particular, our results settle a major part of the conjecture recently put forward in [4].     

A note on Turán type and mean inequalities for the Kummer function

Turán-type inequalities for combinations of Kummer functions involving Φ(a±ν,c±ν,x) and Φ(a,c±ν,x) have been recently investigated in [Á. Baricz, Functional inequalities involving Bessel and modified Bessel functions of the first kind, Expo. Math. 26 (3) (2008) 279-293; M.E.H. Ismail, A. Laforgia, Monotonicity properties of determinants of special functions, Constr. Approx. 26 (2007) 1-9]. In the current paper, we resolve the corresponding Turán-type and closely related mean inequalities for the additional case involving Φ(a±ν,c,x). The application to modeling credit risk is also summarized.     

The maximum size of hypergraphs without generalized 4-cycles

Let fr(n) be the maximum number of edges in an r-uniform hypergraph on n vertices that does not contain four distinct edges A, B, C, D with A∪B=C∪D and A∩B=C∩D=∅. This problem was stated by Erd?s [P. Erd?s, Problems and results in combinatorial analysis, Congr. Numer. 19 (1977) 3-12]. It can be viewed as a generalization of the Turán problem for the 4-cycle to hypergraphs.Let . Füredi [Z. Füredi, Hypergraphs in which all disjoint pairs have distinct unions, Combinatorica 4 (1984) 161-168] observed that ?r?1 and conjectured that this is equality for every r?3. The best known upper bound ?r?3 was proved by Mubayi and Verstraëte [D. Mubayi, J. Verstraëte, A hypergraph extension of the bipartite Turán problem, J. Combin. Theory Ser. A 106 (2004) 237-253]. Here we improve this bound. Namely, we show that for every r?3, and ?₃?13/9. In particular, it follows that ?r→1 as r→∞.     

Differentiation identities for hypergeometric functions

It is well-known that differentiation of hypergeometric function multiplied by a certain power function yields another hypergeometric function with a different set of parameters. Such differentiation identities for hypergeometric functions have been used widely in various fields of applied mathematics and natural sciences. In this expository note, we provide a simple proof of the differentiation identities, which is based only on the definition of the coefficients for the power series expansion of the hypergeometric functions.     

A Turán-type inequality for the gamma function

We prove that the following Turán-type inequality holds for Euler's gamma function. For all odd integers n?1 and real numbers x>0 we have

α?Γ(n−1)(x)Γ(n+1)(x)−Γ(n)²(x),     

On even-cycle-free subgraphs of the hypercube

It is shown that the size of any C4k+2-free subgraph of the hypercube Qn, k?3, is o(e(Qn)).     

Conjectures and theorems in the theory of entire functions

Motivated by the recent solution of Karlin's conjecture, properties of functions in the Laguerre–Pólya class are investigated. The main result of this paper establishes new moment inequalities for a class of entire functions represented by Fourier transforms. The paper concludes with several conjectures and open problems involving the Laguerre–Pólya class and the Riemann -function.     

Vertex Turán problems in the hypercube

Let Qn be the n-dimensional hypercube: the graph with vertex set n{0,1} and edges between vertices that differ in exactly one coordinate. For 1?d?n and F⊆d{0,1} we say that S⊆n{0,1} is F-free if every embedding satisfies i(F)?S. We consider the question of how large S⊆n{0,1} can be if it is F-free. In particular we generalise the main prior result in this area, for F=²{0,1}, due to E.A. Kostochka and prove a local stability result for the structure of near-extremal sets.We also show that the density required to guarantee an embedded copy of at least one of a family of forbidden configurations may be significantly lower than that required to ensure an embedded copy of any individual member of the family.Finally we show that any subset of the n-dimensional hypercube of positive density will contain exponentially many points from some embedded d-dimensional subcube if n is sufficiently large.     

Ramanujan's Eisenstein series and new hypergeometric-like series for

Using hypergeometric identities and certain representations for Eisenstein series, we uniformly derive several new series representations for 1/π².     

On Locally Repeated Values of Certain Arithmetic Functions,IV

Let (n) denote the number of prime divisors of n and let (n) denote the number of prime power divisors of n. We obtain upper bounds for the lengths of the longest intervals below x where (n), respectively (n), remains constant. Similarly we consider the corresponding problems where the numbers (n), respectively (n), are required to be all different on an interval. We show that the number of solutions g(n) to the equation m+(m)=n is an unbounded function of n, thus answering a question posed in an earlier paper in this series. A principal tool is a Turán-Kubilius type inequality for additive functions on arithmetic progressions with a large modulus.     

On Sparse Parity Check Matrices

We consider the extremal problem to determine the maximal number of columns of a 0-1 matrix with rows and at most ones in each column such that each columns are linearly independent modulo . For fixed integers and , we shall prove the probabilistic lower bound = ; for a power of , we prove the upper bound which matches the lower bound for infinitely many values of . We give some explicit constructions.     

Generalized Moments of Additive Functions. II

We obtain upper and lower bounds for the generalized moments of additive arithmetical functions. The results extend relevant theorems proved by I. Z. Ruzsa and A. Hildebrand.     

Turán type inequalities for hypergeometric functions

In this note our aim is to establish a Turán type inequality for Gaussian hypergeometric functions. This result completes the earlier result that G. Gasper proved for Jacobi polynomials. Moreover, at the end of this note we present some open problems.

     

Exact Bounds on the Sizes of Covering Codes

A code c is a covering code of X with radius r if every element of X is within Hamming distance r from at least one codeword from c. The minimum size of such a c is denoted by c _r(X). Answering a question of Hämäläinen et al. [10], we show further connections between Turán theory and constant weight covering codes. Our main tool is the theory of supersaturated hypergraphs. In particular, for n > n ₀(r) we give the exact minimum number of Hamming balls of radius r required to cover a Hamming ball of radius r + 2 in {0, 1}ⁿ. We prove that c _r(B _n(0, r + 2)) = _{1 i r + 1} ( ^{(n + i – 1) / (r + 1) 2}) + n / (r + 1) and that the centers of the covering balls B(x, r) can be obtained by taking all pairs in the parts of an (r + 1)-partition of the n-set and by taking the singletons in one of the parts.     

On non-strong jumping numbers and density structures of hypergraphs

Estimating Turán densities of hypergraphs is believed to be one of the most challenging problems in extremal set theory. The concept of ‘jump’ concerns the distribution of Turán densities. A number α∈[0,1) is a jump for r-uniform graphs if there exists a constant c>0 such that for any family F of r-uniform graphs, if the Turán density of F is greater than α, then the Turán density of F is at least α+c. A fundamental result in extremal graph theory due to Erd?s and Stone implies that every number in [0,1) is a jump for graphs. Erd?s also showed that every number in [0,r!/r^r) is a jump for r-uniform hypergraphs. Furthermore, Frankl and Rödl showed the existence of non-jumps for hypergraphs. Recently, more non-jumps were found in [r!/r^r,1) for r-uniform hypergraphs. But there are still a lot of unknowns regarding jumps for hypergraphs. In this paper, we propose a new but related concept-strong-jump and describe several sequences of non-strong-jumps. It might help us to understand the distribution of Turán densities for hypergraphs better by finding more non-strong-jumps.     

Extension of gamma, beta and hypergeometric functions

The main object of this paper is to present generalizations of gamma, beta and hypergeometric functions. Some recurrence relations, transformation formulas, operation formulas and integral representations are obtained for these new generalizations.     

Some Turán type results on the hypercube

For graphs H,G a classical problem in extremal graph theory asks what proportion of the edges of H a subgraph may contain without containing a copy of G. We prove some new results in the case where H is a hypercube. We use a supersaturation technique of Erd?s and Simonivits to give a characterization of a set of graphs such that asymptotically the answer is the same when G is a member of this set and when G is a hypercube of some fixed dimension. We apply these results to a specific set of subgraphs of the hypercube called Fibonacci cubes. Additionally, we use a coloring argument to prove new asymptotic bounds on this problem for a different set of graphs. Finally we prove a new asymptotic bound for the case where G is the cube of dimension 3.     

Bounds for ratios of modified Bessel functions and associated Turán-type inequalities

New sharp inequalities for the ratios of Bessel functions of consecutive orders are obtained using as main tool the first order difference-differential equations satisfied by these functions; many already known inequalities are also obtainable, and most of them can be either improved or the range of validity extended. It is shown how to generate iteratively upper and lower bounds, which are converging sequences in the case of the I-functions. Few iterations provide simple and effective upper and lower bounds for approximating the ratios Iν(x)/Iν−1(x) and the condition numbers for any x,ν?0; for the ratios Kν(x)/Kν+1(x) the same is possible, but with some restrictions on ν. Using these bounds Turán-type inequalities are established, extending the range of validity of some known inequalities and obtaining new inequalities as well; for instance, it is shown that Kν+1(x)Kν−1(x)/(Kν²(x))<|ν|/(|ν|−1), x>0, ν∉[−1,1] and that the inequality is the best possible; this proves and improves an existing conjecture.