• Korean Journal of Mathematics
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• v.30 no.2
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• pp.199-203
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• 2022

Let f(z) be an entire function of exponential type τ such that ║f║ = 1. Also suppose, in addition, that f(z) ≠ 0 for ℑz > 0 and that $h_f(\frac{\pi}{2})=0$. Then, it was proved by Gardner and Govil [Proc. Amer. Math. Soc., 123(1995), 2757-2761] that for y = ℑz ≤ 0 $${\parallel}D_{\zeta}[f]{\parallel}{\leq}\frac{\tau}{2}({\mid}{\zeta}{\mid}+1)$$, where D_ζ[f] is referred to as polar derivative of entire function f(z) with respect to ζ. In this paper, we prove an inequality in the opposite direction and thereby obtain some known inequalities concerning polynomials and entire functions of exponential type.

• Honam Mathematical Journal
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• v.46 no.2
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• pp.313-334
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• 2024

In the current study, our aim is to define new generalized extended beta and hypergeometric types of functions. Next, we methodically determine several integral representations, Mellin transforms, summation formulas, and recurrence relations. Moreover, we provide log-convexity, Turán type inequality for the generalized extended beta function and differentiation formulas, transformation formulas, differential and difference relations for the generalized extended hypergeometric type functions. Also, we additionally suggest a generating function. Further, we provide the generalized extended beta distribution by making use of the generalized extended beta function as an application to statistics and obtaining variance, coefficient of variation, moment generating function, characteristic function, cumulative distribution function, and cumulative distribution function's complement.