In Mathematics, we see a large number of functions, each having its own properties. Some of these are very interesting and contribute greatly to the intensive research in the field of Mathematics. This paper deals with one such function (which we have termed as the phi function) which emerges from a chain of inequalities, established from the basic concepts of differential calculus. This paper establishes several inequalities which relate to functions and their integrals. Another important expression (from the point of view of notations) links a class of divergent infinite series to the phi function. Finally, we will dive into a brief overview of the phi-form of plane trigonometric functions and derive the trigonometric identity sin2(θ) + cos2(θ) = 1, thus marking their importance. Throughout the paper, we will be analyzing functions in R+ such that the functions are always greater than 0. We will also consider that the functions are continuous and differentiable in the intervals under consideration.