International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 09 Issue: 07 | July 2022 www.irjet.net p-ISSN: 2395-0072 © 2022, IRJET | Impact Factor value: 7.529 | ISO 9001:2008 Certified Journal | Page 2288 Multivariate Analysis of Cauchy’s Inequality Varanasi L V S K B Kasyap 1 , V S Bhagavan 2 , Amrutha Macharla 3 , Devarasetty Syam Sai Akhil 4 1,3,4 Student, School of Computer Science and Engineering, VIT-AP University, Inavolu, India 2 Professor, Dept. of Mathematics, K L Education Foundation, Vaddeswaram, India ---------------------------------------------------------------------***--------------------------------------------------------------------- Abstract – This paper instigates the multivariate generalization of the widely used Cauchy inequality 1 + x ≤ e x where x can be any non-negative real number. The results of this study can be the solution for the Cauchy’s problem for particular Ordinary Differential Equation (ODE). This is also related to study of the complete monotone function and the divided differences theory. The proof is based on the empty product convention notion and the Beppo Levi theorem of Monotone convergence. This study is also extended to multivariate generalization of the simultaneous inequalities. Key Words: ODEs, Inequalities, Population Dynamics, Simultaneous Inequalities, Divided differences. 1.INTRODUCTION The elemental inequality that is being used for various applications in Plant Biology, Olympiad Inequality Problems , Image Processing, Signal processing and various computer application is given in (1). 1 + x ≤ e x (1) Where x can be any non-negative real number, generally x is considered as smaller values in real time applications. However, multivariate generalization of (1) is not established. This is the main focus of this paper is to prove (2) specifically. (2) Where x1, x2……., xn are the pairwise non-negative distinct real numbers. Here ak := The empty product convention is made, so that (2) can be changed as (1) when the n value is 1. In this paper, we also show that the inequality in (2) is only correct when the value of is 0. (1), (2) are extended into whole Euclidean space when n value is 2, and (2) is indeterminate for in the range of (2, -2), (0, -1) and (1, -1/2) or can also be considered as (-1/4, -1/2). From the analogue of specific ODE Cauchy problem, the generalized inequality form (2) is defined. This ODE Cauchy problems are widely used in the plant biology for chromosome analysis, population dynamics, to predict the virus mutations and many other problems. A direct way of solution is required to handle these real-life inequality-based problems. However, this cannot be analysed using the elementary methods since the problems are complex in nature. The monotonic study of functions and mean value theorem of divided difference of functions are essential for complex problems. In next section of this paper ODE approach of solving i.e., the study of Cauchy problem is shown the with the aforementioned solutions. The straightforward approach of preliminary analysis of the given inequality is presented in the section 3 of the paper. The conclusion along with future scope of the work is generalized in the section 4. 2. ODE Approach of solving 2.1 Primary Analysis In this paper, autonomous ODE Cauchy problem is considered as given in (3). (3) Where t1, t2……., tn are the distinct positive real numbers (pairwise) in the increasing order. The convential representations of (3) when n is 1 and n is 2 are the standard logistic model and the standard logistic model with the Alle effect. The eccentric maximal smooth solution can be easily deduced by the concerning classic theory of the Cauchy problems of ODEs from the problems data as (4) and the phase line is depicted in Fig.- 1. where (4) The y in the Fig 1 is,