Trinomial equation: the Hypergeometric way
OMS-Vol. 5 (2021), Issue 1, pp. 236 – 247 Open Access Full-Text PDF
Daniele Ritelli, Giulia Spaletta
Abstract: This paper is devoted to the analytical treatment of trinomial equations of the form \(y^n+y=x,\) where \(y\) is the unknown and \(x\in\mathbb{C}\) is a free parameter. It is well-known that, for degree \(n\geq 5,\) algebraic equations cannot be solved by radicals; nevertheless, roots are described in terms of univariate hypergeometric or elliptic functions. This classical piece of research was founded by Hermite, Kronecker, Birkeland, Mellin and Brioschi, and continued by many other Authors. The approach mostly adopted in recent and less recent papers on this subject (see [1,2] for example) requires the use of power series, following the seminal work of Lagrange [3]. Our intent is to revisit the trinomial equation solvers proposed by the Italian mathematician Davide Besso in the late nineteenth century, in consideration of the fact that, by exploiting computer algebra, these methods take on an applicative and not purely theoretical relevance.
Study of a unit power-logarithmic distribution
OMS-Vol. 5 (2021), Issue 1, pp. 218 – 235 Open Access Full-Text PDF
Christophe Chesneau
Abstract: This article proposes a new unit distribution based on the power-logarithmic scheme. The corresponding cumulative distribution function is defined by a special ratio of power and logarithmic functions that is dependent on one parameter. We show that this function benefits from great flexibility characterized by a large selection of convex and concave shapes. The other key functions are determined and studied. In particular, we show that the probability density function may take on different decreasing or U shapes, and the hazard rate function has a wide panel of U shapes. These functional capabilities are rare for a one-parameter unit distribution. In addition, we prove certain stochastic order results, provide the expression of the quantile function via the Lambert function, some interesting distributional results, and give simple expressions for the ordinary moments, mean, variance, skewness, kurtosis, moment generating function and incomplete moments. Subsequently, a basic statistical approach is described, to show how the new distribution can be applied in a data analysis scenario. Finally, complementary mathematical findings are presented, yielding new integrals linked to the Euler constant via some well-known moments properties.
Starlikeness of meromorphic functions involving certain differential inequalities
OMA-Vol. 5 (2021), Issue 1, pp. 64 – 68 Open Access Full-Text PDF
Kuldeep Kaur Shergill, Sukhwinder Singh Billing
Abstract: In the present paper, we define a class of non-Bazilevic functions in punctured unit disk and study a differential inequality to obtain certain new criteria for starlikeness of meromorphic functions.
On some iterative methods with frozen derivatives for solving equations
OMS-Vol. 5 (2021), Issue 1, pp. 209 – 217 Open Access Full-Text PDF
Samundra Regmi, Christopher Argyros, Ioannis K. Argyros, Santhosh George
Abstract: We determine a radius of convergence for an efficient iterative method with frozen derivatives to solve Banach space defined equations. Our convergence analysis use \(\omega-\) continuity conditions only on the first derivative. Earlier studies have used hypotheses up to the seventh derivative, limiting the applicability of the method. Numerical examples complete the article.
Some Ostrowski type integral inequalities via generalized harmonic convex functions
OMS-Vol. 5 (2021), Issue 1, pp. 200 – 208 Open Access Full-Text PDF
Muhammad Tariq, Saad Ihsan Butt
Abstract: In this paper, we aim to introduce a new notion of convex functions namely the harmonic \(s\)-type convex functions. The refinements of Ostrowski type inequality are investigated which are the generalized and extended variants of the previously known results for harmonic convex functions.
Dye ability of henna dye towards cotton fabrics in comparison with reactive dye by following reactive dyeing procedure
EASL-Vol. 4 (2021), Issue 2, pp. 29 – 35 Open Access Full-Text PDF
Pranay Dutta, Md. Razaya Rabbi, Mohammad Abu Sufian, Shahnaz Mahjebin
Abstract: Although synthetic dyes are commonly used, natural dyes are still being utilized and used to improve their intrinsic aesthetic properties as the main material for the body’s beauty. For example, research results have shown that henna plant leaves comprise dye together with other additives. This provides a hint that if color from henna is properly studied, it can be used not only as body decoration but may also have fiber-substrates affinity. This paper explores the dyeing possibility-the ability to dye and the fastness qualities of henna dye extracted from henna leaves on cotton fabric compared to reactive dyeing using the same dyeing technique as reactive dyeing. Also, color fastness tests have been performed according to the ISO test methods. The implications of henna dye have been shown to have poor to moderate dyeing capability towards cotton fabrics as opposed to the reactive dyes when henna dyeing is accompanied by reactive dyeing. Similarly, henna dye demonstrated satisfactory properties of fastness as opposed to reactive dye. For henna dye with 50% shade, it gives an outstanding color tone with a good level of coloration. Taken into account the ability to dye and the fastness of color, the dyeing of matured henna leaves is equally advantageous to the dyeing of cotton fabrics.
