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Article Citation - WoS: 1Citation - Scopus: 1The Truncated q-bernstein Polynomials in the Case q > 1(Hindawi Ltd, 2014) Turan, MehmetThe truncated q-Bernstein polynomials B-n,B-m,B-q (f; x), n is an element of N, and m is an element of N-0 emerge naturally when the q-Bernstein polynomials of functions vanishing in some neighbourhood of 0 are considered. In this paper, the convergence of the truncated q-polynomials on [0, 1] is studied. To support the theoretical results, some numerical examples are provided.Article Fedja’s Proof of Deepti’s Inequality(Tubitak Scientific & Technological Research Council Turkey, 2018) Ostrovska, Sofiya; Turan, MehmetThe paper aims to present, in a systematic way, an elegant proof of Deepti’s inequality. Both the inequalityand various ideas concerning the issue were discussed on the Mathoverflow website by a number of users, but none haveappeared in the literature thus far. In this work, suggestions pertaining to users ‘Deepti’ and ‘fedja’ are traced, whencethe title. The results or the paper are new, and the proof is divided into a series of statements, many of which are ofinterest in themselves.Article How Analytic Properties of Functions Influence Their Images Under the Limit q-Stancu Operator(Springer Basel AG, 2026) Gurel, Ovgu; Ostrovska, Sofiya; Turan, MehmetIn the study of various q-versions of the Bernstein polynomials, a significant attention is paid to their limit operators. The present work focuses on the impact of the limit q-Stancu operator Sq infinity,alpha on the analytic properties of functions when 0 < q < 1 and alpha > 0. It is shown that for every f is an element of C[0, 1], the function S-q,(alpha infinity)fadmits an analytic continuation into the disk {z : z+alpha/(1-q) < 1+ alpha/(1-q)}. In addition, it is proved that the more derivatives f has at x = 1, the wider this disk becomes. Further, if f is infinitely differentiable at x = 1, then the function S-q,(alpha infinity)fis entire. Finally, some growth estimates for (S-q,(alpha infinity)f)(z) are obtained.Article Citation - WoS: 2Citation - Scopus: 1The Continuity in Q of the Lupaş Q-Analogues of the Bernstein Operators(Academic Press inc Elsevier Science, 2024) Yilmaz, Ovgue Gurel; Turan, Mehmet; Ostrovska, Sofiya; Turan, Mehmet; Ostrovska, Sofiya; Turan, Mehmet; Ostrovska, Sofiya; Mathematics; MathematicsThe Lupas q-analogue Rn,q of the Bernstein operator is the first known q-version of the Bernstein polynomials. It had been proposed by A. Lupas in 1987, but gained the popularity only 20 years later, when q-analogues of classical operators pertinent to the approximation theory became an area of intensive research. In this work, the continuity of operators Rn,q with respect to parameter q in the strong operator topology and in the uniform operator topology has been investigated. The cases when n is fixed and n -> infinity have been considered. (c) 2022 Elsevier Inc. All rights reserved.Article On the Moment-Determinacy of Power Lindley Distribution and Some Applications To Software Metrics(Acad Brasileira de Ciencias, 2021) Khalleefah, Mohammed; Ostrovska, Sofiya; Turan, MehmetThe Lindley distribution and its numerous generalizations are widely used in statistical and engineering practice. Recently, a power transformation of Lindley distribution, called the power Lindley distribution, has been introduced by M. E. Ghitany et at who initiated the investigation of its properties and possible applications. In this article, new results on the power Lindley distribution are presented. The focus of this work is on the moment-(in)determinacy of the distribution for various values of the parameters. Afterwards, certain applications are provided to describe data sets of software metrics.Article Citation - WoS: 4Citation - Scopus: 5Stability Analysis of an Epidemic Model With Vaccination and Time Delay(Wiley, 2023) Turan, Mehmet; Adiguzel, Rezan Sevinik; Koc, F.This paper presents an epidemic model with varying population, incorporating a new vaccination strategy and time delay. It investigates the impact of vaccination with respect to vaccine efficacy and the time required to see the effects, followed by determining how to control the spread of the disease according to the basic reproduction ratio of the disease. Some numerical simulations are provided to illustrate the theoretical results.Article Citation - WoS: 4Citation - Scopus: 4On the Powers of the Kummer Distribution(Academic Publication Council, 2017) Ostrovska, Sofiya; Turan, Mehmet; MathematicsThe Kummer distribution is a probability distribution, whose density is given by f (x) = cx (alpha-1)(1 + delta x)(-gamma) e(-beta x), X > 0, where alpha, beta, delta > 0, gamma is an element of R and C is a normalizing constant. In this paper, the distributions of random variable X-P, p > 0, where X has the Kummer distribution, are considered with the conditions being IFR/DFR, some properties of moments depending on the parameters and the moment-(in) determinacy. In the case of moment-indeterminacy, exemplary Stieltjes classes are constructed.Article Citation - WoS: 1Citation - Scopus: 1Shape-Preserving Properties of the Limit q-durrmeyer Operator(Academic Press inc Elsevier Science, 2024) Yilmaz, Ovgu Gurel; Ostrovska, Sofiya; Turan, MehmetThe present work aims to establish the shape-preserving properties of the limit q- Durrmeyer operator, D q for 0 < q < 1. It has been proved that the operator is monotonicity- and convexity-preserving. What is more, it maps a function m - convex along {q (j)}(infinity)(j =0) to a function m - convex along any sequence { xq( j )}(infinity)(j =0) , x is an element of (0, 1). (c) 2024 Elsevier Inc. All rights reserved.Article Citation - WoS: 1Citation - Scopus: 1On the q-moment Determinacy of Probability Distributions(Malaysian Mathematical Sciences Soc, 2020) Ostrovska, Sofiya; Turan, MehmetGiven 0Article Citation - WoS: 1Citation - Scopus: 1On the Rate of Convergence for the q-durrmeyer Polynomials in Complex Domains(Walter de Gruyter Gmbh, 2024) Gurel, Ovgu; Ostrovska, Sofiya; Turan, MehmetThe q-Durrmeyer polynomials are one of the popular q-versions of the classical operators of approximation theory. They have been studied from different points of view by a number of researchers. The aim of this work is to estimate the rate of convergence for the sequence of the q-Durrmeyer polynomials in the case 0 < q < 1. It is proved that for any compact set D subset of C, the rate of convergence is O(q(n)) as n -> infinity. The sharpness of the obtained result is demonstrated.
