{'fraction_non_alphanumeric': 0.06643869076697606, 'fraction_numerical': 0.02784562774792379, 'mean_word_length': 3.7297921478060045, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Monotonicity Properties of Gaussian Hypergeometric Functions with Respect to the Parameter

The authors establish the necessary and sufficient conditions under which certain combinations of Gaussian hypergeometric function and elementary function are monotone in the parameter, which generalize the recent results of generalized elliptic integrals of the first and second kinds obtained by Qiu et al. Moreover, the authors also prove two monotonicity theorems of generalized elliptic integrals from another point of view.

During the past few years, both complete elliptic integrals (K and E ) and generalized elliptic integrals (K a and E a ) have been widely studied and applied in the theories of conformal invariants, quasiconformal mappings and Ramanujan's modular equations [?, 2-4, 9, 12-14, 17-29, 31, 32]. One of the meaningful tasks is to investigate the dependence on the parameter a in K a and E a , and thus show some stabilities of K a and E a with respect to a and establish several sharp bounds for K a and E a in terms of K and E .
(1) If 1/2 ≤ a < b ≤ 1, then all coefficients are positive in the Taylor series for µ(b) − µ(a) in powers of r 2 .
Let us recall some well-known formulas, which can be found in [1,4].
If f ′ (x)/g ′ (x) is strictly monotone, then the monotonicity in the conclusion is also strict.
Lemma 1.6. ( [16, Lemma 2.1]) For n ∈ N 0 , let r n and s n be real numbers, and let the power series R(x) = ∞ n=0 r n x n and S(x) = ∞ n=0 s n x n be convergent for |x| < 1. If s n ≥ 0 and not all vanish for n ∈ N 0 , and if r n /s n is strictly increasing (decreasing) in n ∈ N 0 , then the function x → R(x)/S(x) is strictly increasing (decreasing, respectively) on (0, 1).
2. The answer to question 1.4 In this section, we always assume that c is a fixed constant in (0, ∞), and a ∈ (0, c/2]. Let 2) 3) For λ ∈ R and |x| < 1, let Now we state our main results below.
Proof of Theorem 2.2 For part (1), by (1.2) and (1.11), we have from which one can easily obtain and therefore and g 1, n is defined in Lemma 2.5.