A family of Horn-Bernstein functions
AA family of Horn-Bernstein functions
Christian Berg and Henrik L. PedersenAugust 17, 2020
Abstract
A family of recently investigated Bernstein functions is revisited andthose functions for which the derivatives are logarithmically completelymonotonic are identified. This leads to the definition of a class of Bernsteinfunctions, which we propose to call Horn-Bernstein functions because ofthe results of Roger A. Horn.
The family of functions h α ( z ) = (1 + 1 /z ) αz := exp( αz Log(1 + 1 /z )) , α ∈ C , (1)defined for z in the cut plane A := C \ ] − ∞ , A → C is the principal logarithm, holomorphic in A and real on thepositive half-line.The functions in (1) appeared in the paper [3] with the goal of finding theset of exponents α > h α is a Bernstein function or equivalently suchthat f α ( x ) = e α − h α ( x ) is a completely monotonic function. This problem wasinspired by a remark in [1, p.458].We adopt the notation of [13] and denote by CM the set of completely mono-tonic functions and BF the set of Bernstein functions. See the monographs [7]and [13] for a treatment of these classes of functions.A family { ϕ α } α ∈ C of entire functions was found in [10, Theorem 2.10] suchthat f α ( z ) = (cid:90) ∞ e − sz ϕ α ( s ) ds, (cid:60) z > . (2)1 a r X i v : . [ m a t h . C A ] A ug hese functions, initially given by a contour integral, were shown to have thepower series expansion ϕ α ( s ) = e α ∞ (cid:88) n =0 ( − n p n +1 ( α ) s n n ! , α, s ∈ C , where ( p n ) n ≥ denotes the sequence of polynomials recursively defined by p ( α ) =1 and p n +1 ( α ) = αn + 1 n (cid:88) k =0 k + 1 k + 2 p n − k ( α ) , n ≥ . (Notice that p ( α ) = α/ p ( α ) = α/ α / f α ∈ CM if and only if ϕ α ( s ) ≥ s >
0. In [10, Theorem 1.7] it was numericallyestablished that there exists a number α ∗ ≈ . ϕ α is non-negative on ]0 , ∞ [ if and only if 0 ≤ α ≤ α ∗ . For α in thisinterval ϕ α is integrable over [0 , ∞ [ and (2) holds for (cid:60) z ≥
0. Furthermore, cf.[10, Theorem 2.11], the Bernstein representation of h α is h α ( z ) = 1 + (cid:90) ∞ (1 − e − sz ) ϕ α ( s ) ds, (cid:60) z ≥ , ≤ α ≤ α ∗ , (4)so the L´evy measure of h α has the density ϕ α with respect to Lebesgue measure.For 0 < α ≤ ϕ α , namely ϕ α ( s ) = 1 π (cid:90) ( x/ (1 − x )) αx sin( απx ) e − sx dx, < α < , s ≥ , and ϕ ( s ) = e − s + 1 π (cid:90) ( x/ (1 − x )) x sin( πx ) e − sx dx, s ≥ . This shows that ϕ α is not only non-negative but in fact that ϕ α ∈ CM . Proofsare given in [10, Section 3].Formula (4) shows that h α is a complete Bernstein function (see [13, Chapter6]) for 0 < α ≤ α = 0. On the other hand f α is not aStieltjes function (defined below) for α > h α ∈ CBF ⇐⇒ α ∈ [0 , . The present paper started as an attempt to find a more direct proof that h ∈ BF , i.e., that h (cid:48) ∈ CM . Our idea was to examine if h (cid:48) is a so-calledlogarithmically completely monotonic function. This is a stronger statementsince the class L of logarithmically completely monotonic functions is definedand characterized by the following result:2 heorem 1.1. The following conditions for a C ∞ -function f : (0 , ∞ ) → (0 , ∞ ) are equivalent and characterize the class L :(i) − (log f ) (cid:48) = − f (cid:48) /f is completely monotonic,(ii) f c is completely monotonic for all c > ,(iii) f /n is completely monotonic for all n = 1 , , . . . . This result goes back to Horn [11]. For more recent proofs and historicalcomments see [2], and [4]. Because of property (ii) we augment the class L to L := L ∪ { } . We define a new class called Horn-Bernstein functions, and denoted by
HBF as HBF := { f ∈ BF | f (cid:48) ∈ L } . Our main result is the following:
Theorem 1.2.
There exists a number ≤ β ∗ < α ∗ such that h α ∈ HBF ⇐⇒ ≤ α ≤ β ∗ . We have β ∗ ≈ . . For a given λ > f : (0 , ∞ ) → R is called a generalized Stieltjesfunction of order λ if f ( x ) = (cid:90) ∞ dµ ( t )( x + t ) λ + c, (5)where µ is a positive measure on [0 , ∞ ) making the integral converge for x > c ≥
0. The set of generalized Stieltjes functions of order λ is denoted S λ .Note that a function in S λ has a holomorphic extension to the cut plane A . Foradditional information on these classes see e.g. [12].The class S of generalized Stieltjes functions of order 1 is just denoted S ,and its members are simply called Stieltjes functions.We remark that f is a generalized Stieltjes function of order λ of the form (5)if and only if f ( x ) = 1Γ( λ ) (cid:90) ∞ e − xt t λ − κ ( t ) dt + c, x > , where κ ∈ CM . In the affirmative case, κ ( t ) = (cid:82) ∞ e − ts dµ ( s ) . See [8, Eqn.(3)] or[12, Lemma 2.1]. This characterization shows also that S λ ⊂ S λ for λ < λ .In accordance with [13, Chapter 8] we define for λ > T BF λ := { f ∈ BF | f (cid:48) ∈ S λ } , m ( s ) such that s − λ m ( s ) is completely monotonic.Clearly T BF λ ⊂ T BF λ for 0 < λ < λ . The class
T BF is also known as the class of Thorin-Bernstein functions simplydenoted T BF , see [13, Chapter 8] and
T BF = CBF (which is easily verified) isthe class of complete Bernstein functions. Since S ⊂ L by [9] we have T BF λ ⊂ CBF ⊂ HBF , < λ < . Concerning the family h α , α ≥ < λ < h α ∈ T BF λ if and only if α = 0.(ii) h α ∈ CBF if and only if 0 ≤ α ≤ h a ∈ HBF if and only if 0 ≤ α ≤ β ∗ .(iv) h a ∈ BF if and only if 0 ≤ α ≤ α ∗ .Only (i) requires a comment. If h α ∈ T BF λ for 0 < λ <
2, then necessarily0 ≤ α ≤
1, and by Equation (4) the L´evy measure for h α has the density ϕ α ,which must satisfy s − λ ϕ α ( s ) ∈ CM . Since ϕ α (0) = e α ( α/
2) is finite, this is onlypossible for α = 0. Remark 1.3.
There is a one-to-one correspondence between the set P ([0 , ∞ [)of infinitely divisible probability measures σ on [0 , ∞ [ and the set of Bernsteinfunctions f satisfying f (0) = 0 via Laplace transformation L ( σ )( x ) := (cid:90) ∞ e − tx dσ ( t ) = e − f ( x ) , x ≥ . Define T λ ⊂ P ([0 , ∞ [) , λ > T λ := { σ ∈ P ([0 , ∞ [) | L ( σ ) = e − f , f ∈ T BF λ , f (0) = 0 } . In [8] there is a discussion of these sets and their relation to exponential families.
A computation shows that h (cid:48) α ( x ) = αh α ( x ) ρ ( x ), where ρ ( x ) = log(1 + 1 /x ) − x + 1 = (cid:90) ∞ e − tx (cid:18) − e − t t − e − t (cid:19) dt. (6)4his function ρ together with the function g defined as g ( x ) = − ρ (cid:48) ( x ) ρ ( x ) (7)will be important in our investigations since h (cid:48) α ∈ L if and only if − h (cid:48)(cid:48) α h (cid:48) α = g − αρ ∈ CM . (8) Remark 2.1.
We notice that ρ ∈ S \ S since (cid:90) t ( x + t ) dt = ρ ( x ) = (cid:90) x + t d ( t − δ ( t )) ,δ denoting the Dirac probability measure with mass at the point 1.The following results are essential ingredients in the search of α for which (8)holds. Proposition 2.2.
The function g in (7) is a Stieltjes function: g ( x ) = 1 x ( x + 1)[( x + 1) log(1 + 1 /x ) −
1] = 1 x + 1 + (cid:90) τ ( t ) x + t dt, (9) where τ is a probability density on ]0 , given by τ ( t ) = (cid:0) t [(1 − t ) log((1 − t ) /t ) − + π t (1 − t ) (cid:1) − . (10) The function τ is convex with a unique minimum at t ∗ ≈ . with minimumvalue m ≈ . . It decreases strictly from ∞ to m on ]0 , t ∗ [ and increases strictlyfrom m to 1 on ] t ∗ , . The function τ roof of Proposition 2.2. We start by noticing that Log(1 + 1 /z ) is holomorphicin C \ [ − ,
0] and that( z + 1) Log(1 + 1 /z ) − (cid:90) − tz + t dt, z ∈ C \ [ − , . In particular ( z + 1) Log(1 + 1 /z ) − (cid:54) = 0 for z ∈ C \ [ − , ρ ( z ) (cid:54) = 0and g ( z ) defined in (7) is holomorphic for z ∈ C \ [ − , g in (9) is easy, and to verify g ∈ S we use that it isequivalent to 1 / ( zg ( z )) ∈ S , cf. [4, p. 25]. However,1 / ( zg ( z )) = ( z + 1)[( z + 1) Log(1 + 1 /z ) −
1] = (cid:90) ( z + 1)(1 − t ) z + t dt = (cid:90) (1 − t ) dt + (cid:90) (1 − t ) z + t dt = 12 + (cid:90) (1 − t ) z + t dt, showing that 1 / ( zg ( z )) ∈ S . Now where we know that g is a Stieltjes function,which is holomorphic in C \ [ − , g ( z ) = c + (cid:90) dτ ( t ) z + t for a positive measure τ on [0 ,
1] and c ≥
0. Since g ( x ) → x → ∞ , we have c = 0. To find the measure τ we use the Stieltjes-Perron inversion formula in theform used in the proof in [1, Lemma 1].For 0 < t < , y > y → + g ( − t + iy ) = {− t (1 − t )[(1 − t )(log((1 − t ) /t ) − iπ ) − } − = − t (1 − t ) { (1 − t ) log((1 − t ) /t ) − − iπ (1 − t ) } − uniformly for t in compact subsets of ]0 , − π lim y → + (cid:61) g ( − t + iy ) = (cid:8) t [(1 − t ) log((1 − t ) /t ) − + π t (1 − t ) (cid:9) − , which shows that τ has the density τ ( t ) given by (10) on the open interval ]0 , τ = m δ + m δ + τ ( t ) dt where m , m ≥ m = lim y → + iyg ( iy ) , m = lim y → + iyg ( − iy ) . We next evaluate the limits as m = 0 , m = 1 thus showing (9).6n fact, for y > iyg ( iy ) = { (1 + iy )[(1 + iy )(Log(1 + iy ) − Log( iy )) − } − = (cid:110) (1 + iy )[(1 + iy )(log (cid:112) y + i arctan y − log y − iπ/ − (cid:111) − , which tends to 0 for y → − log y in the denominator.Similarly iyg ( − iy ) = { ( − iy )[ iy (Log( iy ) − Log( − iy )) − } − = (cid:110) ( − iy )[ iy (log y + iπ/ − log (cid:112) y − ( π − arctan y )) − (cid:111) − , which tends to 1 for y → xg ( x ) → x → ∞ becauselim u → g (1 /u ) /u = lim u → u (1 + u )[(1 + u ) log(1 + u ) − u ] = 2 , by inserting the power series for log(1 + u ). If this is combined with the lastexpression in (9), we get that τ is a probability density.The convexity and monotonicity properties of τ follows e.g. by a Maple pro-gram. (cid:3) A normalized Hausdorff moment sequence is of the form µ n = (cid:90) t n dµ ( t ) , n = 0 , , . . . , where µ is a probability measure on [0 , µ has the density τ ( t ) and τ (1 − t ) with respect to Lebesgue measure on the unit interval will beimportant in the following. Theorem 2.3.
Let ( t n ) denote the Hausdorff moment sequence given by t n := (cid:90) s n τ (1 − s ) ds, n ≥ , (11) and define G α ( x ) := 2 + ∞ (cid:88) n =1 x n n ! (cid:18) t n − αn + 1 (cid:19) , x > . (12) Then h α ∈ HBF ⇐⇒ G α ( x ) ≥ , x > .Proof. From (6) and Proposition 2.2 it follows that g ( x ) − αρ ( x ) = (cid:90) ∞ e − tx F α ( t ) dt, F α ( t ) = e − t + (cid:90) e − ts τ ( s ) ds − α (cid:18) − e − t t − e − t (cid:19) so by (8) and Bernstein’s theorem h (cid:48) α ∈ L if and only if F α ( t ) ≥ t > e t F α ( t ) = 1 + α + (cid:90) e ts τ (1 − s ) ds − α e t − t , so inserting the power series for e ts and e t and using the moments ( t n ) from (11),we get that F α ≥ G α ≥ (cid:3) Remark 2.4.
For x > G α ( x ) is a decreasing function of α , so if G β ( x ) ≥ x > α < β , then also G α ( x ) ≥ x >
0. This means thatthere is a number β ∗ ≤ α ∗ such that h (cid:48) α ∈ L for 0 ≤ α ≤ β ∗ . Here α ∗ is thenumber from (3). Remark 2.5.
Defining the sequence ( a n ) n by a = 1 , a n = 1 /t n − /t n − , n ≥ t n = 1 / ( a + · · · + a n ) . Motivated by [5, Theorem 1.1] we are interested in knowing if ( a n ) n is a Hausdorffmoment sequence. If this is true, then ( t n ) n is an infinitely divisible Hausdorffmoment sequence in the sense that ( t cn ) n is a Hausdorff moment sequence for any c >
0, cf. [6, Proposition 4.2]. Numerical experiments suggest that the necessaryand sufficient conditions of Hausdorff for a sequence to be a Hausdorff momentsequence are satisfied for ( a n ) n , but we have not been able to prove this.That ( t n ) n is infinitely divisible is the same as the claim that the probabilitydensity on [0 , ∞ [ given by d ( s ) = τ (1 − e − s ) e − s , s ≥ We aim at showing the nonnegativity of the function G α given by (12) on thepositive half-line. Our approach is to show that the n th coefficient in the powerseries is positive as n becomes large and to do this we shall find an asymptoticlower bound on the moments ( t n ) n . Not all coefficients in the series G α are positiveand it will be necessary for us to treat the first coefficients, the intermediatecoefficients and the tail of the coefficients using different methods.We first describe how to compute the moments ( t n ) n of the probability mea-sure τ (1 − s ) ds analytically, using among other things the moments of the prob-ability measure τ ( s ) ds . We have (cid:90) τ ( t ) x + t dt = ∞ (cid:88) n =0 ( − n x n +1 s n , x > , (13)8here s n = (cid:90) t n τ ( t ) dt, n = 0 , , . . . . Using (9) we can find another expression for the power series (13). For this set x = 1 /u and consider ϕ ( u ) := g (1 /u ) −
11 + 1 /u = u (1 + u )[(1 + u ) log(1 + u ) − u ] − u u . From the power series for log(1 + u ) , | u | < u ) log(1 + u ) − u = ∞ (cid:88) n =0 ( − n ( n + 1)( n + 2) u n +2 , hence ϕ ( u ) = 2 u u (cid:32) ∞ (cid:88) n =0 ( − n n + 1)( n + 2) u n (cid:33) − − u u . (14)Now write (cid:32) ∞ (cid:88) n =0 ( − n n + 1)( n + 2) u n (cid:33) − = ∞ (cid:88) n =0 ρ n u n . Then ρ = 1 , ρ = 1 / ρ n = n − (cid:88) k =0 ρ k − n − − k ( n − k + 1)( n − k + 2) , n ≥ . Notice that any ρ n can be computed using this recursive formula. Returning to(14) we have ϕ ( u ) = u (cid:32) ∞ (cid:88) n =0 ( − n u n (cid:33) (cid:32) ∞ (cid:88) n =1 ρ n u n (cid:33) = u ∞ (cid:88) n =0 ( − n (cid:32) n (cid:88) k =1 ( − k ρ k (cid:33) u n , which compared with (13) shows that s n = 1 + 2 n (cid:88) k =1 ( − k ρ k , n ≥ . The moments ( t n ) n are given in terms of ( s n ) n as t n = n (cid:88) k =0 ( − k (cid:18) nk (cid:19) s k , n = 0 , , . . . , ρ , . . . , ρ n , then thenumbers s , . . . , s n and finally the numbers t , . . . , t n . The first 6 moments are t = 1 , t = 2 / , t = 5 / , t = 67 / , t = 371 / , t = 1465 / . Next we shall find an asymptotic lower bound on the moments of a large classof probability densities.
Lemma 3.1.
Let p be a probability density on ]0 , such that lim t → p ( t ) = ∞ and with moments µ n , n = 0 , , . . . .For any c > the exists n ∈ N such that µ n > cn + 1 , n ≥ n . Proof.
By assumption on p there exists 0 < T < p ( t ) ≥ c + 1 for T < t <
1. We then get µ n = (cid:90) t n p ( t ) dt ≥ ( c + 1) (cid:90) T t n dt = c + 1 n + 1 (1 − T n +1 ) . The last expression is larger than c/ ( n + 1) iff T n +1 < / ( c + 1) which holds for n sufficiently large. Proposition 3.2.
For n ≥ we have t n > n + 1) , (15) and G ( x ) > for x > . In particular h ∈ HBF .Proof. The density τ (1 − t ) is strictly increasing for t ∈ [1 / ,
1[ and has limit ∞ for t →
1. Therefore there is T ∈ ]1 / ,
1[ such that τ (1 − T ) = 2 and for T < σ < t n > τ (1 − σ ) (cid:90) σ t n dt = τ (1 − σ ) n + 1 (1 − σ n +1 ) . The last expression is ≥ / ( n + 1) if and only if n + 1 ≥ log (cid:16) τ (1 − σ ) − τ (1 − σ ) (cid:17) log( σ ) . Choosing σ = 0 .
985 we get that (15) holds for n ≥
57. That the inequality(15) holds for n = 4 , . . . ,
56 is established by a Maple program, based on thecomputational method described above (see Appendix A). By (12) we thereforeget for x > G ( x ) > (cid:88) n =1 x n n ! (cid:18) t n − n + 1 (cid:19) = 2 − x − x − x x > , where the last inequality is easy to check.10 emark 3.3. As is evident from the proof above not all partial sums of thepower series G ( x ) are positive. The first five terms have to be combined toconclude positivity. Proof of Theorem 1.2.
By Theorem 2.3 and Remark 2.4 we know that β ∗ ≤ α ∗ < . ≤ β ∗ .The number β ∗ can be estimated in the following way similar to the proof ofProposition 3.2. Let 1 / < T < τ (1 − T ) = 2 .
3. For
T < σ < t n > τ (1 − σ ) (cid:90) σ t n dt = τ (1 − σ ) n + 1 (1 − σ n +1 ) . The last expression is ≥ . / ( n + 1) if and only if n + 1 ≥ log (cid:16) τ (1 − σ ) − . τ (1 − σ ) (cid:17) log( σ ) . Choosing σ = 0 .
989 we get that t n > . / ( n + 1) for n ≥
71, while a Mapleprogram (similar to the code in Appendix A) shows that it holds for n = 5 , . . . , ≤ α ≤ . P N ( x, α ) := 2 + N (cid:88) n =1 x n n ! (cid:18) t n − αn + 1 (cid:19) , and assume that N ≥
5. Since G α ( x ) > P N ( x, α ) for x > α ≤ β ∗ if P N ( x, α ) is non-negative for x >
0. As an example it can be checked easily by aMaple computation that P N ( x, α ) > N = 20 and α = 2 . G α are less than 1 then G α ( x ) − P N ( x, α ) < R N ( x ) , R N ( x ) = ∞ (cid:88) n = N +1 x n n ! , so if p := P N ( x , α ) is the global minimum of P N ( x, α ) over [0 , ∞ [ assumednegative and if R N ( x ) < | p | , then G a ( x ) < β ∗ < α . As an exampleit can be checked easily by a Maple computation that P N ( x, α ) for N = 20 and α = 2 . p ≈ − . x ≈ . R ( x ) ≈ . − .Using this method β ∗ can easily be estimated very accurately and the com-putations yield the specific number in the theorem. (cid:3) Let us end the paper by mentioning that β ∗ can be determined as the minimumvalue of a certain function M defined in the following proposition. The graph of(approximations to) this function indicates again the minimum value as β ∗ .11 roposition 3.4. The function M ( x ) := 2 + (cid:80) ∞ n =1 t n n ! x n (cid:80) ∞ n =1 x n ( n +1)! , x > is positive, continuous and tends to infinity for x tending to 0 and to infinity. Itsminimum over ]0 , ∞ [ equals β ∗ .Proof. It is clear that M tends to infinity for x → c > n be as in Lemma 3.1 when p ( t ) = τ (1 − t ). Setting R c ( x ) = ∞ (cid:88) n = n x n ( n + 1)! , we find M ( x ) ≥ (cid:80) n − n =1 t n n ! x n + cR c ( x ) (cid:80) n − n =1 x n ( n +1)! + R c ( x ) , hence lim inf x →∞ M ( x ) ≥ c . Since c > x →∞ M ( x ) = ∞ . The minimum value is clearly the largest number α > G α ≥ A Maple code
For the reader’s convenience we have included the Maple code used in the proofof Proposition 3.2.
N:=56; [0]:=1;for n to N dot[n] := sum((-1)^k*binomial(n,k)*s[k], k = 1 .. n) + s[0];end do References [1] H. Alzer and C. Berg, Some classes of completely monotonic functions,
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