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**11 - 16**of**16**### Complete monotonicity of a polygamma function plus the square of another polygamma function, submitted

"... Abstract. For m, n ∈ N, let ..."

### TWO NEW PROOFS OF THE COMPLETE MONOTONICITY OF A FUNCTION INVOLVING THE PSI FUNCTION

, 902

"... Abstract. In the present paper, we give two new proofs for the necessary and sufficient condition α ≤ 1 such that the function x α [lnx − ψ(x)] is completely monotonic on (0, ∞). 1. ..."

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Abstract. In the present paper, we give two new proofs for the necessary and sufficient condition α ≤ 1 such that the function x α [lnx − ψ(x)] is completely monotonic on (0, ∞). 1.

### INEQUALITIES FOR THE GAMMA FUNCTION AND ESTIMATES FOR THE VOLUME OF SECTIONS OF B n p

, 2000

"... Abstract. Let Bn p = {(xi) ∈ Rn; ∑n 1 |xi | p ≤ 1} and let E be a k-dimensional subspace of Rn. We prove that |E ∩ Bn p | 1/k k ≥ |Bn p | 1/n n, for 1 ≤ k ≤ (n − 1)/2 and k = n − 1 whenever 1 < p < 2. We also consider 0 < p < 1 and other related cases. We obtain sharp inequalities invol ..."

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Abstract. Let Bn p = {(xi) ∈ Rn; ∑n 1 |xi | p ≤ 1} and let E be a k-dimensional subspace of Rn. We prove that |E ∩ Bn p | 1/k k ≥ |Bn p | 1/n n, for 1 ≤ k ≤ (n − 1)/2 and k = n − 1 whenever 1 < p < 2. We also consider 0 < p < 1 and other related cases. We obtain sharp inequalities involving Gamma function in order to get these results.

### ANZIAM J. 44(2003), 609–623 INEQUALITIES FOR THE BETA FUNCTION OF n VARIABLES

, 2001

"... We present various inequalities for Euler’s beta function of n variables. One of our theorems states that the inequalities an 1Qn iD1 xi − B.x1; : : : ; xn / bn () hold for all xi 1 (i D 1; : : : ; n; n 3) with the best possible constants an D 0 and bn D 1 − 1=.n − 1/W. This extends a recently ..."

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We present various inequalities for Euler’s beta function of n variables. One of our theorems states that the inequalities an 1Qn iD1 xi − B.x1; : : : ; xn / bn () hold for all xi 1 (i D 1; : : : ; n; n 3) with the best possible constants an D 0 and bn D 1 − 1=.n − 1/W. This extends a recently published result of Dragomir et al., who investigated () for the special case n D 2. 1.