Volume 72, Issue 5

In this paper, we mainly show that generalized hyperharmonic number sums with reciprocal binomial coefficients can be expressed in terms of classical (alternating) Euler sums, zeta values and generalized (alternating) harmonic numbers.

The Gini index of a set partition π of size n is defined as 1−δ(π)n2, $1-\frac{\delta (\pi )}{{{n}^{2}}},$where δ ( π ) is the sum of the squares of the block cardinalities of π . In this paper, we study the distribution of the δ statistic on various kinds of set partitions in which the first r elements are required to lie in distinct blocks. In particular, we derive the generating function for the distribution of δ on a generalized class of r -partitions wherein contents-ordered blocks are allowed and elements meeting certain restrictions may be colored. As a consequence, we obtain simple explicit formulas for the average δ value, equivalently for the average Gini index, in all r -partitions, r -permutations and r -Lah distributions of a given size. Finally, combinatorial proofs can be found for these formulas in the case r = 0 corresponding to the Gini index on classical set partitions, permutations and Lah distributions.

Let f k ( n ) := ∑ d k ∣ n Φ ( d ) , $$ f_k(n):=\sum\limits_{d^k \mid n} \Phi(d), $$ where Φ( d ) is a multiplicative function, Φ( d ) = O ( d ε ). We study asymptotic behaviour of the sum T f k c ( N ) := ∑ n ≤ N f k n c , 1 < c < 2 , $$ T_{f_k}^c(N):=\sum\limits_{n \leq N} f_k\left(\left\lfloor n^c\right\rfloor\right), \quad 1<c<2, $$ and establish some asymptotic formulas for T f k c ( N ) $ T_{f_k}^c(N) $under assumptions on f k .

In this paper, we present some generalizations and further refinements of Young-type inequality due to Choi [Math. Inequal. Appl. 21 (2018), 99–106], which strengthen the results obtained by Ighachane et al. [Math. Inequal. Appl. 23 (2020), 1079–1085]. As applications of these scalars results, we can get some inequalities for determinants, trace and p -norms of τ -measurable operators.

In this paper, our aim is to prove certain kinds of Steffensen type integral inequalities for the pseudo-integral and the discrete pseudo-integral. The observations concern two cases of the real semiring with pseudo-operations with respect to pseudo-integrals: the first semiring, where pseudo-operations are defined via a monotone and continuous function g , the second semiring, when pseudo-operations are given by an idempotent addition and a generated pseudo-multiplication. Moreover, the discrete pseudo-integral is based on symmetric pseudo-addition and pseudo-multiplication, where the generator g is odd and increasing. In each case, several practical examples are presented to illustrate these results.

Let A be the set of all analytic functions f defined on the open unit disk D satisfying f (0) = f' (0) − 1 = 0. Let φ Ne ( z ) := 1 + z − z 3 = 3 be the recently introduced Carathéodory function which maps the unit circle ∂ D $ \partial \mathbb{D} $D onto a 2-cusped kidney-shaped curve called nephroid given by ((u−1)2+v2−4a)3−4v22=0. ${{\left( {{(u-1)}^{2}}+{{v}^{2}}-\frac{4}{a} \right)}^{3}}-\frac{4{{v}^{2}}}{2}=0.$In this paper, we determine the best possible estimate on the real β so that for some analytic p satisfying p (0) = 1 the following subordination-implication holds: 1+βzp′(z)pj(z)≺F(z)⇒p(z)≺φNe(z), j=0,1,2, $$1+\beta \frac{z{{p}^{\prime }}(z)}{{{p}^{j}}(z)}\prec \mathcal{F}(z)\Rightarrow p(z)\prec {{\varphi }_{Ne}}(z),\quad j=0,1,2,$$ where F ( z ) is some Carathéodory function with special geometries like right/left-half of Bernoulliφs lemniscate, cardioid, lune, eight-shaped, etc. As applications, we establish sufficient conditions for the Ma-Minda family of nephroid starlike functions given by SNe*:={ f∈A:zf′(z)f(z)≺φNe(z) }. $$\mathcal{S}_{Ne}^{*}:=\left\{ f\in \mathcal{A}:\frac{z{{f}^{\prime }}(z)}{f(z)}\prec {{\varphi }_{Ne}}(z) \right\}.$$

In this paper, we study the following linear differential system (1) x′(t)=A(t)x(t),    x(t)∈ℝn, t∈ℝ, $${{x}^{\prime }}(t)=A(t)x(t),\,\,\,\,x(t)\in {{\mathbb{R}}^{n}},\quad t\in \mathbb{R},$$ where t ↦ A ( t ) is a matrix valued almost periodic function. We prove that if all the solutions of the above system are almost periodic, there exists an almost periodic function b : R → R n such that the following differential equation (2) x′(t)=A(t)x(t)+b(t),    x(t)∈ℝn, t∈ℝ $${{x}^{\prime }}(t)=A(t)x(t)+b(t),\,\,\,\,x(t)\in {{\mathbb{R}}^{n}},\quad t\in \mathbb{R}$$ has no bounded solution. In particular, if for each almost periodic function b there exists a bounded solution to (2), there exists at least one solution for (1) that is not almost periodic.

This paper deals with the existence and numerical estimates of solutions for a class of fractional differential equations, while the nonlinear part of the problem admits some Special hypotheses. In particular, for a precise localization of the parameter, the existence of a non-zero solution is established requiring the sublinearity of nonlinear part at origin and infinity. Moreover, theoretical and numerical examples of applications are provided.

In this present work, we perceive the ideas of rough weighted statistical limit set as well as rough weighted statistical cluster points set and originate these conceptions into asymmetric metric spaces . On this context we frame out several results which substantially intensify these perceptions. While explicating such notions in terms of their asymmetric concepts, this generalization despite unfollows some previous results rather generates new characteristics. Also, we will adorn a sufficient condition using rough weighted statistical convergence which converts asymmetric metric spaces to approximate metric spaces.

Balázs operators are an influential tool that can be used to approximate a function on the unbounded interval [0, ∞). In this study, a new construction of Balázs operators which depending upon on a function ρ ( x ) has been introduced. The function ρ ( x ) plays a significant role in this construction due to the fact that the new operator preserves definitely two test functions from the set of {1, ρ ( x ), ρ 2 ( x )}. In this direction, the approximation properties of this newly defined operators are established, such as degree of approximation and Voronovskaya type theorems. Finally, we presented a set of computational examples in order to validate the new construction of operator is an approximation procedure.

In this paper, η -biharmonic hypersurfaces with constant scalar curvature in 5-dimensional pseudo-Riemannian space forms are studied. We prove that such hypersurfaces with diagonalizable shape operator have constant mean curvature, which gives an affirmative partial answer to the conjecture in [Arvanitoyeorgos, A.—Kaimakamis, F. G.: Hypersurfaces of type M23 $\begin{array}{} \displaystyle M^3_2 \end{array}$ in E24 $\begin{array}{} \displaystyle \mathbb{E}^4_2 \end{array}$ with proper mean curvature vector, J. Geom. Phys. 63 (2013), 99–106]. As a result, we give several partial classification results.

In this paper, we prove that every hemi-slant warped product submanifold of the form N θ × f N ⊥ in a nearly trans-Sasakian manifold M͠ satisfies the following inequality: ∥ h ∥ 2 ≥ n 2 cot 2 θ (∥∇̂(ln f )∥ 2 – β 2 ), whereas the warped product by reversing these two factors, i.e., N ⊥ × f N θ satisfying the inequality: ∥ h∥ 2≥ n19cos2⁡ θ (∥ ∇ ^ (ln⁡ f)∥ 2− β 2), $\begin{array}{} \displaystyle \|h\|^2\geq \frac{n_1}{9}\cos^2\theta(\|\widehat\nabla(\ln f)\|^2-\beta^2), \end{array}$ where n 1 = dim N θ , n 2 = dim N ⊥ , ∇̂(ln f ) is the gradient of ln f and ∥ h ∥ is the length of the second fundamental form of the warped product immersion in M͠ . The equality cases of these inequalities are investigated. Furthermore, we discuss some special cases of these inequalities. Finally, we construct two non-trivial examples.

The symbol 𝓕 n ( X ) denotes the hyperspace of all nonempty subsets of a Hausdorff space X having at most n points. This hyperspace is endowed with the Vietoris topology. For a mapping between Hausdorff spaces f : X → Y , define the induced mapping f n : 𝓕 n ( X ) → 𝓕 n ( Y ) by f n ( A ) = f ( A ) (the image of A under f ). In this paper, we study the relationship between the condition f belongs to a class of mappings between Hausdorff spaces 𝕄 and the condition f n belongs to 𝕄.

This study introduces a parsimonious and tractable generator for continuous distribution called the Teissier-G family of distributions for continuous random variables and examines the distributions belonging to this family as the sub-models. Some general statistical characteristics and sub-models of the new generator were examined and studied. Similarly, we examined the shapes of the sub-models probability density function (pdf) and hazard rate function were investigated. The parameter of the proposed model was obtained in a closed form by maximum likelihood. In addition to the numerical real life applications, Monte Carlo simulation was performed to examine the flexibility of the introduced models. The models provide good fits in all the cases. The results show great improvement compared to existing models.

In this paper, we introduced a new extension of the beta generator of distributions. Some important properties of the model are discussed, such as the series representation, quantile function, moments, Rényi entropy, and order statistics. Three special members are discussed, namely, the new extended beta exponential (NEBE), new extended beta uniform (NEBU), and new extended beta half logistic (NEBH). The model parameters estimation was achieved by the method of maximum likelihood and the Bayes estimation under square error loss function (SEL); the estimators were examined by simulation studies and it was a very good result as discussed by the bias and mean square error (MSE) of the estimators. Two real data applications are provided to illustrate the importance of the new model in practice.

The logarithmic distribution and a given lifetime distribution are compounded to construct a new family of lifetime distributions. The compounding is performed with respect to maxima. Expressions are derived for lifetime properties like moments and the behavior of extreme values. Estimation procedures for the method of maximum likelihood are also derived and their performance assessed by a simulation study. Three real data (including two lifetime data) applications are described that show superior performance (assessed with respect to Kolmogorov Smirnov statistics, likelihood values, AIC values, BIC values, probability-probability plots and density plots) versus at least five known lifetime models, with each model having the same number of parameters as the model it is compared to.

A parametric regression model may comprise several correlated individual regression equations, and these equations may have common and different unknown parameters. In such a situation, the common unknown parameter vectors in these equations can be estimated individually or simultaneously according to various available statistical inference methods. The purpose of this paper is to provide an integral account of two classic objects in regression theory: the best linear unbiased predictors (BLUPs) and the best linear unbiased estimators (BLUEs) on common and different unknown parameters in two correlated linear models with common and different parameters. We first introduce some reduced models associated with the two correlated linear models. We then define and characterize predictability and estimability of all unknown parameter vectors in the two correlated models and their reduced models, and derive analytical formulas for calculating the BLUPs and BLUEs of all unknown parameter vectors in these models by means of a constrained quadratic matrix optimization method. We also discuss a variety of theoretical properties of the predictors and estimators.

Let 𝔻 = { z ∈ C : | z | < 1} be the unit disk and Hol(𝔻 × 𝔻) be the space of all holomorphic functions on the bi-disc 𝔻 × 𝔻. We consider the double convolution operator 𝒦 f on the subspace Hol zw (𝔻 × 𝔻) := { f ∈ Hol(𝔻 × 𝔻) : f ( z,w ) = g ( zw ) for some g ∈ Hol(𝔻)} defined by K f h ( z w ) = ( f ∗ h ) ( z w ) := ∫ 0 z ∫ 0 w f ( ( z − u ) ( w − v ) ) h ( u v ) d v   d u . $$\left( {{\mathcal{K}}_{f}}h \right)(zw)=(f*h)(zw):=\int\limits_{0}^{z}{\int\limits_{0}^{w}{f}}((z-u)(w-v))h(uv)\text{d}v~\text{d}u.$$ We study extended eigenvalues of 𝒦 f . We characterize extended eigen vectors of 𝒦 f in terms of Duhamel operators. Moreover, we describe cyclic vectors of operator 𝒦 f by applying the Duhamel product method.