Abstract In the present paper, we study triangular numbers. We focus on the linear homogeneous recurrence relation of degree 3 with constant coefficients for triangular numbers. Then we deal with the relationship between generalized Fibonacci polynomials and triangular numbers. We show that different properties of triangular numbers can be obtained by using this relationship. Finally, we examine the properties of the sequence A052529 that has strong relationships with triangular numbers.

Abstract For a zero-dimensional Hausdorff space X, denote, as usual, by C(X, ℤ) the ring of continuous integer-valued functions on X. If f ∈ C(X, ℤ), denote by Z(f) the set of all points of X that are mapped to 0 by f. The set $$\begin{array}{} \displaystyle C_K(X,\mathbb Z)=\{f\in C(X,\mathbb Z)\mid \text{cl}_X(X\smallsetminus \mathsf Z(f))\text{ is compact}\} \end{array}$$ is the integer-valued analogue of the ideal of functions with compact support in C(X). By first working in the category of locales and then interpreting the results in spaces, we characterize this ideal in several ways. Writing ζ X for the Banaschewski compactification of X, we also explore some properties of ideals of C(X, ℤ) associated with subspaces of ζ X analogously to how one associates, for any Tychonoff space Y, subsets of β Y with ideals of C(Y).

Abstract 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.

Abstract 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 φNe (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 + v 2 − 4 a ) 3 − 4 v 2 2 = 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 + β z p ′ ( z ) p j ( z ) ≺ F ( z ) ⇒ p ( z ) ≺ φ N e ( 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 S N e * : = { f ∈ A : z f ′ ( z ) f ( z ) ≺ φ N e ( z ) } . $$\mathcal{S}_{Ne}^{*}:=\left\{ f\in \mathcal{A}:\frac{z{{f}^{\prime }}(z)}{f(z)}\prec {{\varphi }_{Ne}}(z) \right\}.$$

Abstract 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 2cot2 θ(∥∇̂(ln f)∥2 – β 2), whereas the warped product by reversing these two factors, i.e., N ⊥ × f N θ satisfying the inequality: $\begin{array}{} \displaystyle \|h\|^2\geq \frac{n_1}{9}\cos^2\theta(\|\widehat\nabla(\ln f)\|^2-\beta^2), \end{array}$ 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.

Abstract In this paper, we first propose a new general method to introduce various lifetime distributions by choosing an appropriate kernel distribution. They have some characteristics in common with the well-known Birnbaum-Saunders distribution. Then, we choose the triangular distribution as a kernel model and construct the new distribution. This distribution has its support on the positive real axis and consists of two-pieces. We show that the newly defined distribution is in fact a generalized Birnbaum-Saunders distribution. It is mathematically tractable for studying its theoretical properties in detail. Different methods of estimation of parameters are proposed. The existence and uniqueness problem of the maximum likelihood estimation method is discussed. The performances of the estimators are evaluated through simulation studies. A real data fitting which compares it with the ordinary Birnbaum-Saunders, Laplace Birnbaum-Saunders and other some generalized Birnbaum-Saunders distributions is also given.

Abstract In this paper, we investigate the solvability of a fourth-order differential evolution equation on singular cylindrical domain containing a cuspidal point. Some regularity results are obtained for the classical solutions by using the Dunford operational calculus.

Abstract With the aid of convolution theorem of the Laplace transforms, a monotonicity rule for the ratio of two Laplace transforms, Bernstein’s theorem for completely monotonic functions, and other analytic techniques, the author presents decreasing property of a ratio constituted via three derivatives of a sum involving trigamma function and discovers necessary and sufficient conditions for a function constituted via three derivatives of a function involving trigamma function to be completely monotonic.

Abstract We shall prove a variety of results on certain types of Putcha semigroups such as: left weakly commutative semigroups, 𝓡-commutative semigroups, 𝓡𝓒-commutative semigroups, generalized conditionally commutative semigroups.

Abstract In this paper, under some conditions on weighted function K, we give some new equivalent characterizations of inner functions in Q K spaces. Meanwhile, we also studied inner functions in Q K spaces as multipliers of C 𝓑(Q K ) ∩ BMOA, where C 𝓑(Q K ) denoted the closure of Q K spaces in Bloch spaces.