Therefore, for follows from 2. If , then from Figure 1 we know that could be positive or negative. We divide the proof into two cases. If , then from 2. Then the properties of and lead to. Hence is strictly decreasing on and strictly increasing on. Therefore, it follows from 2.

Thus is strictly decreasing on and strictly increasing on. By the same argument as Case 1, for follows from 2. If and , then from 2.

### Submission history

Thus there exists such that for. Since as , we know that there exists such that for , which implies that is strictly increasing on and. Therefore, for and for. We state a simple lemma as the results of [ 12 , 47 ]. For , it follows from 2. If , then it follows from 3. Therefore, is strictly logarithmically completely monotonic on that follows from 3. Conversely, if , then we can divide the set into two subsets: and.

## On a Class of Incomplete Gamma Functions with Applications

Therefore, it follows from Lemma 2. If , then from 3. For , it follows from 3. Conversely, if and , then we can divide the set into two subsets: and.

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Although the upper and lower bounds of Kershaw's inequalities given in 1. For example,. Furthermore, the advantage of our inequalities is to give the upper and lower bounds of Kershaw's inequality for and while Laforgia established only one side of Kershaw's inequality.

Korean Mathematical Society , 25 3 — Journal of Inequalities and Applications , International Journal of Modern Mathematics , 3 1 — Bulletin of the Institute of Mathematics. Academia Sinica , 3 3 — International Journal of Pure and Applied Mathematics , 38 4 — Chen Ch-P: Monotonicity and convexity for the gamma function. Journal of Inequalities in Pure and Applied Mathematics , 6 4 :6, article no. Bulletin of the Korean Mathematical Society , 47 1 — Integral Transforms and Special Functions , 21 1—2 — Qi F: A completely monotonic function involving the divided difference of the psi function and an equivalent inequality involving sums.

Journal of the Indonesian Mathematical Society , 11 1 — Taiwanese Journal of Mathematics , 7 2 — Qi F: Monotonicity results and inequalities for the gamma and incomplete gamma functions.

## On a Class of Incomplete Gamma Functions with Applications - CRC Press Book

Alzer H: Some gamma function inequalities. Mathematics of Computation , 60 — Alzer H: On some inequalities for the gamma and psi functions. Mathematics of Computation , 66 — Proceedings of the American Mathematical Society , 11 — Kershaw D: Some extensions of W. Gautschi's inequalities for the gamma function.

- On an extension of generalized incomplete Gamma functions with applications;
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Mathematics of Computation , 41 — Merkle M: Logarithmic convexity and inequalities for the gamma function. Journal of Mathematical Analysis and Applications , 2 — Alzer H, Berg Ch: Some classes of completely monotonic functions. The Ramanujan Journal , 11 2 — Alzer H: Sharp inequalities for the digamma and polygamma functions. Forum Mathematicum , 16 2 — Alzer H, Batir N: Monotonicity properties of the gamma function.

Applied Mathematics Letters , 20 7 — Analysis and Applications , 1 1 — Proceedings of the American Mathematical Society , 9 — Mathematics of Computation , 47 — Journal of Mathematical Analysis and Applications , 1 :1—9. Ukrainian Mathematical Journal , 59 10 — Muldoon ME: Some monotonicity properties and characterizations of the gamma function.

Aequationes Mathematicae , 18 1—2 — Integral Transforms and Special Functions , 17 7 — Journal of Mathematical Analysis and Approximation Theory , 1 1 — Integral Transforms and Special Functions , 18 5—6 — Qi F: A class of logarithmically completely monotonic functions and the best bounds in the first Kershaw's double inequality.

Journal of Computational and Applied Mathematics , 2 — Journal of Mathematical Analysis and Applications , 1 — Computational Intelligence Foundations and Applications , 4: — Chen Ch-P: Complete monotonicity properties for a ratio of gamma functions. Univerzitet u Beogradu.

Serija Matematika , 26— Merkle M: On log-convexity of a ratio of gamma functions. Serija Matematika , 8: — Journal of the Korean Mathematical Society , 45 1 — Journal of the Korean Mathematical Society , 45 2 — Laforgia A: Further inequalities for the gamma function. The following tables give peak and mean relative errors in over various domains of a and z, along with comparisons to the GSL Note that only results for the widest floating point type on the system are given as narrower types have effectively zero error.

### Additional information

Note also that the higher error rates for the 80 and bit long double results are somewhat misleading: expected results that are zero at bit double precision may be non-zero - but exceptionally small - with the larger exponent range of a long double. These results therefore reflect the more extreme nature of the tests conducted for these types.

There are two sets of tests: spot tests compare values taken from Mathworld's online evaluator with this implementation to perform a basic "sanity check". Accuracy tests use data generated at very high precision using NTL's RR class set at bit precision using this implementation with a very high precision term Lanczos approximation , and some but not all of the special case handling disabled.

This is less than satisfactory: an independent method should really be used, but apparently a complete lack of such methods are available.

go site We can't even use a deliberately naive implementation without special case handling since Legendre's continued fraction see below is unstable for small a and z. These four functions share a common implementation since they are all related via:. However there is another series representation for the lower integral:. That lends itself to calculation of the upper integral via rearrangement to:. Refer to the documentation for powm1 and tgamma1pm1 for details of their implementation.

Note however that the precision of tgamma1pm1 is capped to either around 35 digits, or to that of the Lanczos approximation associated with type T - if there is one - whichever of the two is the greater. That therefore imposes a similar limit on the precision of this function in this region.

There are two special cases used when a is an integer or half integer, and the crossover conditions listed above indicate that we should compute the upper integral Q. While for half integers in the range 0.

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- On a Class of Incomplete Gamma Functions with Applications - PDF Free Download.

These are both more stable and more efficient than the continued fraction alternative. In this area an expansion due to Temme is used:. The double sum is truncated to a fixed number of terms - to give a specific target precision - and evaluated as a polynomial-of-polynomials. There are versions for up to bit long double precision: types requiring greater precision than that do not use these expansions. The coefficients C k n are computed in advance using the recurrence relations given by Temme.