Completely monotone sequences and harmonic mappings
aa r X i v : . [ m a t h . C V ] J a n COMPLETELY MONOTONE SEQUENCES AND HARMONICMAPPINGS
BO-YONG LONG, TOSHIYUKI SUGAWA, AND QI-HAN WANG
Abstract.
In the present paper, we will study geometric properties of harmonic map-pings whose analytic and co-analytic parts are (shifted) generated functions of completelymonotone sequences. Introduction and preliminaries
A sequence { a n } ∞ n =0 of real numbers is called completely monotone (or totally monotone )if ∆ k a n ≥ n, k ≥ . Here, ∆ k a n is defined recursively by ∆ a n = a n , n ≥ , and ∆ k a n := ∆ k − a n − ∆ k − a n +1 , n ≥ , k ≥ . Note that a completely monotone sequence is non-negative, non-increasing and convex.Hausdorff [9] showed that { a n } is a completely monotone sequence precisely when thereis a positive Borel measure µ on [0 ,
1] such that a n = Z t n dµ ( t ) , n ≥ . (1.1)Therefore, the word “completely monotone sequence” is a synonym of “Hausdorff momentsequence”. When a = 1 the sequence is said to be normalized . Note that the condition a = 1 means that µ is a probability measure. In particular, the generating function ofthe normalized Hausdorff sequence { a n } is represented in the form F ( z ) = 1 + ∞ X n =1 a n z n = Z dµ ( t )1 − tz (1.2)for a Borel probability measure µ on [0 , . We denote by T the set of those functions F generated by normalized Hausdorff moment sequences. For instance, letting µ be theDirac measure with unit mass at t = 0 or t = 1 , we see that the functions F ( z ) = 1 and F ( z ) = 1 / (1 − z ) belong to T . By the form (1.2), we observe that a function F ∈ T is analytically continued to the slit domain Λ := C \ [1 , + ∞ ). We note that F ( x ) is non-decreasing in −∞ < x < F ( x ) has a limit (possibly + ∞ ) as Mathematics Subject Classification.
Primary 31A05, 30E05; Secondary 30C62, 44A60.
Key words and phrases.
Harmonic mappings; completely monotone sequences; Hausdorff momentsequences; quasiconformal mappings.The present research was supported in part by Natural Science Foundation of Anhui Province(1908085MA18), Foundation of Anhui Educational Committee (KJ2020A0002), China. x → − . The value of the limit will be denoted by F (1 − ) . By the form of F, we observethat(1.3) | F ( z ) | ≤ ∞ X n =0 a n | z | n = F ( | z | ) ≤ F (1 − ) = ∞ X n =0 a n , | z | < . Note also that F (1 − ) ≥ a = 1 . We denote by e T the set of shifted generated functions zF ( z ) for F ∈ T . For instance, the functions f ( z ) = z and f ( z ) = z/ (1 − z ) both aremembers of e T . Completely monotone sequences are closely related with moment problems and thetheory of continued fractions and thus important not only in analysis but also in proba-bility and applied mathematics, see [13, 22, 26] for instance. In recent years, the theoryof universally prestarlike functions (containing universally convex and universally starlikefunctions) was developed and an intimate connection with T was found (see [20] and[21]). As is well recognized, many kinds of special functions may be described in terms offunctions in T (see [3], [21] as well as Section 4 below).Let Hol(Λ) denote the set of analytic functions on the domain Λ = C \ [1 , + ∞ ) . Thefollowing lemma is more or less known to experts. This sort of result was formulated in[21, Lemma 2.1] and then simplified by Liu and Pego [13] (see Remark 2 therein).
Lemma 1.1.
Let F ∈ Hol(Λ) . Then F ∈ T , i.e. F can be represented in the form F ( z ) = Z dµ ( t )1 − tz for a Borel probability measure µ on [0 , , if and only if the following three conditions arefulfilled: (i) F (0) = 1 ; (ii) F ( x ) is a non-negative real number for each x ∈ ( −∞ , ; (iii) Im F ( z ) ≥ whenever Im z > .Moreover, the measures µ and the functions F are in one-to-one correspondence. Let Har( D ) denote the class of complex-valued harmonic functions on the unit disk D . Then, each function f in Har( D ) is uniquely expanded in the form(1.4) f ( z ) = ∞ X n = −∞ a n r | n | e inθ , z = re iθ ∈ D . For another F ( z ) = P n A n r | n | e inθ in Har( D ) , we define the (harmonic) convolution (orthe Hadamard product ) of f and F by( f ∗ F )( z ) = ∞ X n = −∞ a n A n r | n | e inθ . Note that f ∗ F ∈ Har( D ) whenever f, F ∈ Har( D ) . It is often more convenient to express f in (1.4) in the form f ( z ) = ∞ X n =1 a − n ¯ z n + ∞ X n =0 a n z n = g ( z ) + h ( z ) , OMPLETELY MONOTONE SEQUENCES AND HARMONIC MAPPINGS 3 where h ( z ) = ∞ X n =0 a n z n and g ( z ) = ∞ X n =1 b n z n ( b n = a − n , n ≥ . The analytic functions h and g are called the analytic part and the co-analytic part of f, respectively. The convolution of harmonic functions f = h + ¯ g and f = h + ¯ g isdescribed also by f ∗ f = h ∗ h + g ∗ g , where h ∗ h and g ∗ g are the ordinary Hadamard products. See [19] for basics ofconvolutions of anaytic functions.A smooth map f : D → C is locally univalent at z if the Jacobian J f = | f z | − | f ¯ z | does not vanish at z by the Inverse Mapping Theorem. Lewy’s theorem asserts that theconverse is true for harmonic mappings. Therefore, a harmonic mapping f = h + ¯ g islocally univalent and sense-preserving at z if and only if J f ( z ) = | f z ( z ) | − | f ¯ z ( z ) | = | h ′ ( z ) | − | g ′ ( z ) | > . In particular, then we have h ′ ( z ) = 0 and the function ω f ( z ) = g ′ ( z ) h ′ ( z )is holomorphic at z and satisfies the inequality | ω f ( z ) | < . We denote by S H the set of sense-preserving harmonic univalent functions f in Har( D )normalized by f (0) = f z (0) − . In what follows, we will mean sense-preserving andinjective (one-to-one) by the term “univalent”. Set also S = { f ∈ S H : f ¯ z (0) = 0 } . These classes were introduced and studied by Clunie and Sheil-Small [5]. Nowadays,many researchers are studying them and their subclasses intensively. See the monograph[7] for fundamental theory and recent progress of harmonic univalent mappings. Notethat ω f = ¯ f z /f z = g ′ /h ′ satisfies the inequality | ω f | < D for f = h + ¯ g ∈ S H . Thequantity ω f = ¯ f z /f z = f ¯ z /f z is called the second complex dilatation of f. If f is univalentand if | ω f | ≤ k for a constant k < , the mapping f is called k -quasiconformal (or K -quasiconformal in the most of the literature, where K = (1 + k ) / (1 − k )). For the theoryof quasiconformal mappings, the reader should consult the standard monograph [1] byAhlfors. Much attention has been paid to the class of harmonic quasiconformal mappingson the unit disk. See [4, 10, 15, 25] and references therein.For a constant c with | c | < , we define the class HT ( c ) to be the set of functions f ∈ Har(Λ) of the form f ( z ) = h ( z ) + cg ( z )for some h, g ∈ e T . In other words, each member f of HT ( c ) is represented as f ( z ) = Z z − tz dµ ( t ) + c Z z − t ¯ z dν ( t )for Borel probability measures µ, ν on [0 , . The purpose of this article is to study geo-metric properties of functions in HT ( c ) such as univalence, convexity in one direction,and quasiconformality.In the next section, main results of this paper will be presented. Their proofs are givenin Section 3. In Section 4, we will give a couple of examples and apply some of the mainresults to polylogarithms and shifted hypergeometric functions. B.-Y. LONG, T. SUGAWA, AND Q.-H. WANG Main results
Theorem 2.1.
Let c be a real constant with ≤ c < . For f ∈ HT ( c ) and a constant a ≥ , the following inequality holds: | a + f ( z ) | ≥ a + f ( −| z | ) ≥ a + lim r → − f ( − r ) , z ∈ D . We remark that the above inequality is meaningful only when a + f ( −| z | ) > . Inparticular, if a + lim r → − f ( − r ) ≥ f is non-constant, then we conclude that a + f ( z ) is non-vanishing on | z | < . If c = 0, this theorem reduces to the main result of[16].Let H = { z : Re z < } . Then we get the following result. Theorem 2.2.
Let f = h + c ¯ g ∈ HT ( c ) for a real constant c with ≤ c < . Suppose h ( z ) = R z (1 − tz ) − dµ ( t ) and g ( z ) = R z (1 − tz ) − dν ( t ) for Borel probability measures µ and ν on [0 , . Then for z = x + iy ∈ H with y = 0 , unless f is a constant function,the following hold: (i) y ∂∂y Re f ( z ) < ; (ii) y ∂∂x Im f ( z ) > provided that µ = cν + (1 − c ) λ for a Borel probability measure λ on [0 , . Wirths [27] proved the following useful result.
Lemma 2.3.
Each function h ∈ e T is univalent on H = { z : Re z < } and the imagedomain D = h ( H ) is convex in the direction of the imaginary axis. Here and hereafter, a domain D in C is said to be convex in the direction of theimaginary axis if the intersection of D with each line parallel to the imaginary axis isconnected (or empty). We can extend this result to the harmonic case. Theorem 2.4.
Let c be a real constant with ≤ c < and f ∈ HT ( c ) . If f is locallyunivalent on H = { z : Re z < } , then f is univalent on H and the image f ( H ) is convexin the direction of the imaginary axis. Unfortunately, we cannot drop the local univalence of f in the assumption. See Example4.2 in Section 4.A linear combination is an important method to construct a new function, cf. [14, 25].However, it is well known that the convex combination of two univalent analytic functionsis not necessarily univalent, let alone convex combination of two univalent harmonic map-pings. The harmonic convolution f ∗ f of two harmonic functions f and f in Har( D )does not necessarily enjoy properties of f or f , such as convexity or even (local) univa-lence (see [6] for instance). However, the following proposition shows that the harmonicconvolution and convex combinations keep the family HT ( c ) invariant in some sense. Proposition 2.5.
Let c , c be complex constants with | c j | < . Then, for f j ∈ HT ( c j ) , j =1 , , the following hold: (i) sf + (1 − s ) f ∈ HT ( c ) for ≤ s ≤ if c = c = c ; OMPLETELY MONOTONE SEQUENCES AND HARMONIC MAPPINGS 5 (ii) f ∗ f ∈ HT ( c c ) . The next result gives us a sufficient condition for a function in HT ( c ) with a constant | c | < D . Theorem 2.6.
Let h ∈ e T and let c be a real constant with ≤ c < . Suppose that (cid:12)(cid:12)(cid:12)(cid:12) h ′ ( tz ) h ′ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ M, z ∈ D , ≤ t ≤ . (2.1) Then for each g ∈ e T , the harmonic mapping f ( z ) := h ( z ) + c ( h ∗ g )( z ) , z ∈ D (2.2) belongs to HT ( c ) and is k -quasiconformal if cM ≤ k < . Note that h ′ ( z ) has no zeros on D since h ∈ e T is univalent on H by Lemma 2.3. Letting t = 0 , we observe that the condition | h ′ ( z ) | ≥ /M, z ∈ D , is necessary for (2.1). However,it is not easy to check (2.1) in general. The following result may be helpful to find a valueof M. Proposition 2.7.
Let h be an analytic function on D with h ′ (0) = 1 and let m be apositive constant. Suppose that the following inequality holds: (2.3) Re zh ′′ ( z ) h ′ ( z ) > − m, z ∈ D . Then (cid:12)(cid:12)(cid:12)(cid:12) h ′ ( tz ) h ′ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ e m , z ∈ D , ≤ t ≤ . It is well known that a normalized analytic function h ( z ) = z + a z + . . . maps D univalently onto a convex domain if and only if Re [ zh ′′ ( z ) /h ′ ( z )] ≥ − . Therefore, wecan take 1 as the constant m in the above proposition for this h. Let h, g ∈ e T . It is important to look at the quotient of the derivatives of two functionsin e T when considering the local univalence or the quasiconformality of the function of HT ( c ) for a consant c . Under what conditions g ′ /h ′ belong to T ? This question isinteresting in itself. In this context the following result proves to be useful. Theorem 2.8.
Let h, g ∈ e T be represented by h ( z ) = Z zφ ( t )1 − tz dt, g ( z ) = Z zψ ( t )1 − tz dt (2.4) for nonnegative Borel functions φ and ψ on (0 , with R φ ( t ) dt = R ψ ( t ) dt = 1 . If theinequality φ ( s ) ψ ( t ) ≥ φ ( t ) ψ ( s )(2.5) holds for < s ≤ t < , then g/h and g ′ /h ′ both belong to T . Note that the claim g/h ∈ T was first proved in [21, Theorem 1.10]. When φ is non-vanishing, the condition in (2.5) means that the function ψ ( t ) /φ ( t ) is non-decreasing in0 < t < . B.-Y. LONG, T. SUGAWA, AND Q.-H. WANG
Using Theorem 2.8, we obtain another sufficient condition for a function in HT ( c ) tobe quasiconformal. Theorem 2.9.
Under the hypotheses of Theorem 2.8, further assume that the func-tion F ( z ) = g ′ ( z ) /h ′ ( z ) has a finite limit F (1 − ) . Then the function f = h + c ¯ g is k -quasiconformal on D if ≤ cF (1 − ) ≤ k < . If lim x → − g ′ ( x ) = g ′ (1 − ) < + ∞ , then we have F (1 − ) = g ′ (1 − ) /h ′ (1 − ) . If g ′ (1 − ) = + ∞ , then h ′ (1 − ) = + ∞ by the assumption F (1 − ) < + ∞ . In this case, we may use l’Hˆospital’srule if the right-most limit below exists: F (1 − ) = lim x → − g ′ ( x ) h ′ ( x ) = lim x → − g ′′ ( x ) h ′′ ( x ) . Proofs of the main results
In this section, we prove all the results in the previous section.
Proof of Theorem 2.1.
By assumption, f = h + cg for some g, h ∈ e T . We first note theinequalities for z = x + iy with r = | z | < ≤ t ≤ z − tz = x − tr − tx + t r ≥ − r − tr tr + t r = − r tr ≥ −
11 + t because the function x ( x − tr ) / (1 − tx + t r ) is increasing in − r ≤ x ≤ r. Letting µ and ν be the representing measures of h and g, respectively, we therefore have theestimates for z with | z | = r < | a + f ( z ) | ≥ a + Re f ( z ) = a + Re h ( z ) + c Re g ( z ) = a + Re h ( z ) + c Re g ( z )= a + Z Re (cid:18) z − tz (cid:19) dµ ( t ) + k Z Re (cid:18) z − tz (cid:19) dν ( t ) ≥ a + Z − r tr dµ ( t ) + c Z − r tr dν ( t )= a + h ( − r ) + cg ( − r ) = a + h ( − r ) + cg ( − r ) = a + f ( − r ) ≥ a + lim r → − f ( − r ) . (cid:3) Proof of Theorem 2.2.
For z = x + yi, by a straightforward computation, we have theexpression Re f ( z ) = Z x − t ( x + y )1 − xt + t ( x + y ) ( dµ ( t ) + cdν ( t )) andIm f ( z ) = Z y − xt + t ( x + y ) ( dµ ( t ) − cdν ( t )) . OMPLETELY MONOTONE SEQUENCES AND HARMONIC MAPPINGS 7
Therefore, we have ∂∂y Re f ( z ) = Z − yt (1 − xt )(1 − xt + t ( x + y )) ( dµ ( t ) + cdν ( t )) , and ∂∂x Im f ( z ) = Z yt (1 − xt )(1 − xt + t ( x + y )) ( dµ ( t ) − cdν ( t )) . Since µ + cν and µ − cν = (1 − c ) λ are positive measures, we have the required inequalitiesfor x < y = 0 . (cid:3) For the proof of Theorem 2.4, we need to recall the shear construction developed byClunie and Sheil-Small [5]. The following form is a vertical version of a theorem of Clunieand Sheil-Small [5, Theorem 5.3] (see also [7, p. 37]) .
Lemma 3.1 (Clunie and Sheil-Small) . Let f = h + ¯ g be a locally univalent harmonicmapping on D . Then f maps D univalently onto a convex domain in the direction of theimaginary axis if and only if the analytic function F = h + g maps D univalently onto aconvex domain in the direction of the imaginary axis. We denote by f ∗ the π/ if of f about the origin. Then f = h + ¯ g is convexin the direction of the imaginary axis if and only if f ∗ = if = ih + i ¯ g = h ∗ − g ∗ is convexin the direction of the real axis. Therefore, the above version follows from the originalversion [5, Theorem 5.3]. Proof of Theorem 2.4.
We will combine the technique employed by Wirths [27] with theshear construction. First note that the M¨obius transformation ϕ ( ζ ) = 2 ζ ζ maps D onto H . Suppose that f = h + c ¯ g ∈ HT ( c ) for some 0 ≤ c < F = h + cg. Then F/ (1 + c ) ∈ e T and the proof of Wirth’s theorem (Lemma 2.3) in [27] now impliesthat F = F ◦ ϕ = h ◦ ϕ + cg ◦ ϕ is univalent on D and convex in the direction of theimaginary axis. Since f := f ◦ ϕ = h ◦ ϕ + cg ◦ ϕ is locally univalent by assumption, nowLemma 3.1 implies that f is univalent on D and convex in the direction of the imaginaryaxis. Since f = f ◦ ϕ − , the assertion now follows. (cid:3) To prove the second part of Proposition 2.5, we need the following lemma.
Lemma 3.2.
Let f, g ∈ e T . Then f ∗ g ∈ e T . This fact is known to experts (see Roth, Ruscheweyh and Salinas [18, p. 3172]). Letus, however, give a proof because the authors could not find a proof in the literature.
Proof.
Let f ( z ) = P a n z n +1 and g ( z ) = P b n z n +1 for normalized completely monotonesequences { a n } and { b n } . We have to show that { a n b n } is completely monotone, too. Wefirst note the formula ∆ k ( a n b n ) = k X j =0 (cid:18) kj (cid:19) ∆ k − j a n + j · ∆ j b n B.-Y. LONG, T. SUGAWA, AND Q.-H. WANG for n, k ≥ . This can be shown by induction on k with the simple identities ∆( A n B n ) =(∆ A n ) B n + A n +1 ∆ B n and (cid:0) kj (cid:1) + (cid:0) kj − (cid:1) = (cid:0) k +1 j (cid:1) for 1 ≤ j ≤ k. Since ∆ k − j a n + j ≥ j b n ≥ , we obtain ∆ k ( a n b n ) ≥ . (cid:3) This result is also claimed by Reza and Zhang [17, Lemma 1.9]. According to them,this follows from the fact that { a n b n } corresponds to the convolution measure µ ⋄ ν when { a n } and { b n } correspond to measures µ and ν, respectively.We are now ready to prove Proposition 2.5. Proof of Proposition 2.5.
Let f j = h j + c j ¯ g j for j = 1 , h j ( z ) = Z − tz dµ j ( t ) and g j ( z ) = Z − tz dν j ( t )for some Borel probability measures µ j , ν j for j = 1 , . The first assertion immediatelyfollows from the fact that (1 − s ) µ + sµ and (1 − s ) ν + sν are Borel probability measuresfor 0 ≤ s ≤ . For the second assertion, we express f ∗ f in the form( f ∗ f )( z ) = ( h ∗ h )( z ) + c c ( g ∗ g )( z ) . By Lemma 3.2, we have h ∗ h , g ∗ g ∈ e T . Thus the assertion follows. (cid:3)
Proof of Theorem 2.6.
Since h, g ∈ e T , by Lemma 3.2, h ∗ g ∈ e T . Thus, it is easy to seethat the function f given in (2.2) belongs to HT ( c ) . Next we prove the quasiconformality of f . Since h, g ∈ e T , h and g can be expressed as h ( z ) = z ∞ X n =0 a n z n , g ( z ) = z ∞ X n =0 b n z n , (3.1)for some Hausdorff moment sequences { a n } and { b n } . Furthermore, there exists a Borelprobability measure ν on [0 ,
1] such that b n = Z t n dν ( t ) , n = 0 , , , . . . . (3.2)A simple computation leads to( h ∗ g )( z ) = ∞ X n =0 a n b n z n +1 = ∞ X n =0 (cid:18) a n z n +1 Z t n dν ( t ) (cid:19) = Z ∞ X n =0 a n z n +1 t n ! dν ( t ) = Z h ( tz ) t dν ( t ) . We remark that this property indeed characterizes the generating functions of Hausdorffmoment sequences (see Grinshpan [8, Theorem 1]). Thus( h ∗ g ) ′ ( z ) = Z h ′ ( tz ) dν ( t ) . OMPLETELY MONOTONE SEQUENCES AND HARMONIC MAPPINGS 9 and therefore ω f = ¯ f z f z = c ( h ∗ g ) ′ h ′ = c Z h ′ ( tz ) h ′ ( z ) dν ( t ) . By the assumption (2.1), we have | ω f ( z ) | = c (cid:12)(cid:12)(cid:12)(cid:12)Z h ′ ( tz ) h ′ ( z ) dν ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c Z (cid:12)(cid:12)(cid:12)(cid:12) h ′ ( tz ) h ′ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) dν ( t ) ≤ cM Z dν ( t ) = cM ≤ k < . In particular, f is locally univalent and thus, by Theorem 2.4, f is univalent on D . Since f is smooth on D , the inequality | ω f | ≤ k < f is k -quasiconformal on D . (cid:3) Proof of Proposition 2.7.
First we note that h ′ ( z ) vanishes nowhere on D by assumption.Let u ( z ) = Re zh ′′ ( z ) h ′ ( z )for z ∈ D . Then u is harmonic and u > − m on D and u (0) = 0 . If we put U = ( u + m ) /m, then U > U (0) = 1 . Thus the Harnack inequality implies the inequality U ( z ) ≥ (1 − r ) / (1 + r ) for | z | = r < . Hence,(3.3) u ( z ) = mU ( z ) − m ≥ − mr r , r = | z | < . Next we set ψ ( s ) = log | h ′ ( sz ) | = Re log h ′ ( sz ) , ≤ s ≤ , for a fixed z ∈ D . Then, by (3.3), ψ ′ ( s ) = Re zh ′′ ( sz ) h ′ ( sz ) = u ( sz ) s ≥ − ms | z | s (1 + s | z | ) ≥ − m. An integration of the above inequality in t ≤ s ≤ | h ′ ( z ) || h ′ ( tz ) | = ψ (1) − ψ ( t ) = Z t ψ ′ ( s ) ds ≥ − m (1 − t ) ≥ − m, which yields the required inequality. (cid:3) Proof of Theorem 2.8.
Since the assertion g/h ∈ e T and its proof are contained in [21],we only show the assertion F := g ′ /h ′ ∈ e T . Indeed, we will employ the same method asin [21].It suffices to check the three conditions in Lemma 1.1 for F. By the expressions in (2.4),we have h ′ ( z ) = Z φ ( t )(1 − tz ) dt and g ′ ( z ) = Z ψ ( t )(1 − tz ) dt. In particular, for a real number x < , we have h ′ ( x ) ≥ g ′ ( x ) ≥ F ( z ) = h ′ ( z ) g ′ ( z ) − h ′ ( z ) g ′ ( z )2 i | h ′ ( z ) | , we have only to show that ( h ′ g ′ − h ′ g ′ ) /i is non-negative on the upper half-plane Im z > . We now compute h ′ ( z ) g ′ ( z ) = Z φ ( s )(1 − sz ) dt Z ψ ( t )(1 − tz ) dt = Z Z φ ( s ) ψ ( t )(1 − sz ) (1 − tz ) dsdt = Z Z s ≤ t + Z Z t ≤ s = Z Z s ≤ t (cid:18) φ ( s ) ψ ( t )(1 − sz ) (1 − tz ) + φ ( t ) ψ ( s )(1 − tz ) (1 − sz ) (cid:19) dsdt = Z Z s ≤ t φ ( s ) ψ ( t )(1 − tz ) (1 − sz ) + φ ( t ) ψ ( s )(1 − sz ) (1 − tz ) | − sz | | − tz | dsdt. Taking the complex conjugate, we have similarly h ′ ( z ) g ′ ( z ) = Z Z s ≤ t φ ( s ) ψ ( t )(1 − sz ) (1 − tz ) + φ ( t ) ψ ( s )(1 − tz ) (1 − sz ) | − sz | | − tz | dsdt. We obtain h ′ ( z ) g ′ ( z ) − h ′ ( z ) g ′ ( z )= 4 i Z Z s ≤ t y ( t − s ) { − ( s + t ) x + str }{ φ ( s ) ψ ( t ) − φ ( t ) ψ ( s ) }| − sz | | − tz | dsdt, where z = x + iy, r = | z | . Since1 − ( s + t ) x + str ≥ − ( s + t ) x + stx = (1 − sx )(1 − tx ) ≥ (1 − s )(1 − t ) ≥ x ∈ ( −∞ , , condition (iii) is now easily confirmed as required. (cid:3) Proof of Theorem 2.9.
By hypothesis, ω f = ¯ f z /f z = cg ′ /h ′ = cF. Note that h ′ is non-vanishing on D by Lemma 2.3. The inequality (1.3) now leads to | ω f | ≤ cF (1 − ) ≤ k < . In particular, f is locally univalent and thus Theorem 2.4 implies that f is univalent on D . We now conclude that f is k -quasiconformal on D . (cid:3) OMPLETELY MONOTONE SEQUENCES AND HARMONIC MAPPINGS 11 Examples and applications
Let us first see an explicit estimate of the constant M in Theorem 2.6. Example 4.1.
Let h ( z ) = z/ (1 − z ). Then h ∈ e T as we remarked in Introduction. Since h ′ ( z ) = 1 / (1 − z ) , we have for z = x + iy with fixed r = | z | < , (cid:12)(cid:12)(cid:12)(cid:12) h ′ ( tz ) h ′ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) (1 − z ) (1 − tz ) (cid:12)(cid:12)(cid:12)(cid:12) = 1 − x + r − tx + t r ≤ (1 + r ) (1 + tr ) < t ) . Hence, sup z ∈ D , ≤ t ≤ (cid:12)(cid:12)(cid:12)(cid:12) h ′ ( tz ) h ′ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = 4 . Taking M = 4 in Theorem 2.6, we know that f given in (2.2) is 4 c -quasiconformal on D for 0 ≤ c < / . We next give a simple example to examine the conditions in Theorems 2.4 and 2.6.
Example 4.2.
Let h ( z ) = z/ (1 − z ) as above and g ( z ) = z . Note that h, g ∈ e T . For apositive constant c < f ( z ) = h ( z ) + cg ( z ) = z/ (1 − z ) + c ¯ z in HT ( c ) . Note that h ∗ g = g in this case. The previous example tells us that f is4 c -quasiconformal on D for c < / . This bound is sharp. Indeed, the second complexdilatation of f is ω f ( z ) = cg ′ ( z ) /h ′ ( z ) = c (1 − z ) and thus satisfies k ω f k ∞ = 4 c. Moreover, f is not locally univalent on D for each c > / . We will show it. Let γ be the intersectionof the circle | z − | = 1 / √ c and D . Note that γ is non-empty because 1 / √ c < . Pointsin this arc γ may be parametrized as z = 1 + e iθ / √ c. Then f (cid:0) e iθ / √ c (cid:1) = 1 + e iθ / √ c − e iθ / √ c + c (1 + e − iθ / √ c ) = c − , which shows that the open arc γ shrinks to the one point c − . Therefore, f is not locallyunivalent at each point of γ. Let us now take a look at polylogarithms. The polylogarithmic function of order α isdefined by Li α ( z ) = ∞ X n =1 z n n α , z ∈ D , α ≥ . By the well-known representationLi α ( z ) = z Γ( α ) Z ( − log t ) α − − tz dt for α > , and Li ( z ) = z/ (1 − z ) , we see that Li α ∈ e T for α ≥ . Also the relationLi α Li β ∈ T , ≤ α ≤ β, (4.1)follows from Theorem 2.8 and was already contained in [21, Lemma 5.1]. Lewis [12] provedthat Li α maps D univalently onto a convex domain for each α ≥ . Furthermore, in [21],the polylogarithmic function Li α is shown to be universally starlike for α = 0 and 1 ≤ α and universally convex for α = 0 , ≤ α and was conjectured to be universally starlike also for 0 < α < < α < . This conjecture wascompletely proved by Bakan, Ruscheweyh and Salinas [2].We need the following estimate below. Though this is essentially contained in [21], weinclude a direct proof of it for convenience of the reader.
Lemma 4.3.
Let F ∈ T and µ be its representing measure on [0 , . Then the followinginequalities hold: Re F ( z ) ≥ Z dµ ( t )1 + t ≥ , z ∈ D . Proof.
By assumption, F is expressed by F ( z ) = Z − tz dµ ( t ) , z ∈ Λ . Letting z = x + iy and r = | z | < , we computeRe F ( z ) = Z − tx − tx + t r dµ ( t ) . Since the function x (1 − tx ) / (1 − tx + t r ) is increasing in − r ≤ x ≤ r for fixed r and t, we have the estimatesRe F ( z ) ≥ F ( − r ) = Z tr tr + t r dµ ( t ) = Z
11 + tr dµ ( t ) ≥ Z
11 + t dµ ( t ) . (cid:3) We apply the above observations to polylogarithms to have the following.
Theorem 4.4.
Let α, β ≥ and c be a non-negative real constant and set f = Li α + c Li β . (i) f is k -quasiconformal on D when α ≤ β and c ≤ k < f is k -quasiconformal on D when < β ≤ α and c ζ ( β − /ζ ( α − ≤ k < . Here, ζ ( s ) denotes the Riemann zeta function s ∞ X n =1 n − s . Recall that ζ ( s ) < + ∞ for s ∈ (1 , + ∞ ) . Proof.
Put h = Li α and g = Li β for brevity. Note that they are univalent on D by Lewis’theorem [12]. Since zh ′ ( z ) = Li α − ( z ) , zg ′ ( z ) = Li β − ( z ) we have F := g ′ h ′ = zg ′ zh ′ = Li β − Li α − . First assume that α ≤ β. Then G := 1 /F ∈ T by (4.1). Hence, Lemma 4.3 implies | G ( z ) | ≥ Re G ( z ) ≥ / z ∈ D . In view of the form of f, we now estimate | ω f | = (cid:12)(cid:12)(cid:12)(cid:12) ¯ f z f z (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) cg ′ h ′ (cid:12)(cid:12)(cid:12)(cid:12) = c | G | ≤ c ≤ k < D . In particular, f is locally univalent and thus, by Theorem 2.4, f is univalent on D . It is now clear that f is k -quasiconformal on D . OMPLETELY MONOTONE SEQUENCES AND HARMONIC MAPPINGS 13
Next assume that α ≥ β. Then F ∈ T by (4.1). If β > , we have F (1 − ) =Li β − (1 − ) / Li α − (1 − ) = ζ ( β − /ζ ( α − < + ∞ . Now the assertion follows from Theorem2.9. (cid:3)
We remark that we have F (1 − ) = + ∞ when 0 ≤ β ≤ β < α. Finally, we apply our results to hypergeometric functions. We recall the definition ofthe hypergeometric function F ( a, b ; c ; z ): F ( a, b ; c ; z ) = 1 + ∞ X n =1 ( a ) n ( b ) n ( c ) n n ! z n , | z | < , where ( a ) n is the Pochhammer symbol; namely, ( a ) n = a ( a + 1) · · · ( a + n −
1) for n ≥ a ) = 1 . Here, a, b and c are (possibly complex) parameters with c = 0 , − , − , . . . . Geometric properties such as starlikeness and convexity of F ( a, b ; c ; z ) and the shifted one z F ( a, b ; c ; z ) were studied by many authors (see [11], [23], [24] and references therein).In particular, in connection with the class T , some observations on the hypergeometricfunctions were made in [21]. The formula(4.2) F ( a, b ; c ; 1 − ) = Γ( c )Γ( c − a − b )Γ( c − a )Γ( c − b ) , Re ( c − a − b ) > , is due to Gauss. It should also be note that the derivatve formula ddz F ( a, b ; c ; z ) = abc F ( a + 1 , b + 1; c + 1; z ) holds. The well-known Euler representation formula F ( a, b ; c ; z ) = Γ( c )Γ( a )Γ( c − a ) Z (1 − tz ) − b t a − (1 − t ) c − a − dt for Re c > Re a > L a,c ( z ) = z F ( a, c ; z )belongs to the class e T for real parameters c > a > µ given by dµ ( t ) = Γ( c )Γ( a )Γ( c − a ) t a − (1 − t ) c − a − dt. We note that L a,c ( z ) is univalent on the half-plane Re z < L ′ a,c is non-vanishing there. The convolution f ∗ L a,c with analytic functions f is oftencalled the Carlson-Shaffer operator and studied by many authors.As a simple application of Theorem 2.8, we have the following result. Lemma 4.5.
Let a, c, a ′ , c ′ be real constants with c > a > and c ′ > a ′ > . If a ′ ≥ a andif c − a ≥ c ′ − a ′ , then the functions L a ′ ,c ′ /L a,c and L ′ a ′ ,c ′ /L ′ a,c both belong to the class T . In a similar way to Theorem 4.4, we finally obtain the following.
Theorem 4.6.
Let a, c, a ′ , c ′ be real constants with c > a > and c ′ > a ′ > . For anon-negative real constant b, set f = L a,c + b L a ′ ,c ′ . (i) f is k -quasiconformal on D when a ≥ a ′ , c − a ≤ c ′ − a ′ and b ≤ k < (ii) f is k -quasiconformal on D when a ′ ≥ a, < c ′ − a ′ ≤ c − a and bM ≤ k < , where M = ( c ′ − c ′ − c − a − c − a − c − c − c ′ − a ′ − c ′ − a ′ − . Proof.
Put h = L a,c and g = L a ′ ,c ′ for brevity. First assume that a ≥ a ′ , c − a ≤ c ′ − a ′ . Then by the previous lemma, G = h ′ /g ′ ∈ T . Therefore, Lemma 4.3 implies | G | ≥ Re G ≥ / D . We estimate as before | ω f | = (cid:12)(cid:12)(cid:12)(cid:12) bg ′ h ′ (cid:12)(cid:12)(cid:12)(cid:12) = b | G | ≤ b ≤ k < D and thus conclude that f is k -quasiconformal on D . Next assume that a ′ ≥ a and 2 < c ′ − a ′ ≤ c − a. Then, by Lemma 4.5, we have F = g ′ /h ′ ∈ T . Note that h ′ ( z ) = F ( a, c ; z ) + ( a/c ) z F ( a + 1 , c + 1; z ) . By (4.2)and the basic identity Γ( x + 1) = x Γ( x ) ,h ′ (1 − ) = Γ( c )Γ( c − a − c − a )Γ( c −
1) + ac · Γ( c + 1)Γ( c − a − c − a )Γ( c − c − c − a − ac · c ( c − c − a − c − a −
2) = ( c − c − c − a − c − a − . Similarly, we have g ′ (1 − ) = ( c ′ − c ′ − c ′ − a ′ − c ′ − a ′ − . Hence, F (1 − ) = M < + ∞ . Now the assertion follows from Theorem 2.9. (cid:3)
References
1. L. V. Ahlfors,
Lectures on Quasiconformal Mappings , second ed., University Lecture Series, vol. 38,American Mathematical Society, Providence, RI, 2006, With supplemental chapters by C. J. Earle,I. Kra, M. Shishikura and J. H. Hubbard.2. A. Bakan, St. Ruscheweyh, and L. Salinas,
Universal convexity and universal starlikeness of polylog-arithms , Proc. Amer. Math. Soc. (2015), 717–729.3. A. Bakan, St. Ruscheweyh, and L. Salinas,
On geometric properties of the generating function for theRamanujan sequence , Ramanujan J. (2018), 173–188.4. S.-L. Chen and S. Ponnusamy, Radial length, radial John disks and K -quasiconformal harmonicmappings , Potential Anal. (2019), 415–437.5. J. Clunie and T. Sheil-Small, Harmonic univalent functions , Ann. Acad. Sci. Fenn. Ser. A I Math. (1984), 3–25.6. M. Dorff, M. Nowak, and M. Wo loszkiewicz, Convolutions of harmonic convex mappings , ComplexVar. Elliptic Equ. (2012), 489–503.7. P. Duren, Harmonic Mappings in the Plane , Cambridge Tracts in Mathematics, vol. 156, CambridgeUniversity Press, Cambridge, 2004.8. A. Z. Grinshpan,
Hausdorff ’s moment sequences and exponential convolutions , Methods Appl. Anal. (1996), 31–45.9. F. Hausdorff, Summationsmethoden und Momentfolgen. I , Math. Z. (1921), 74–109.10. D. Kalaj, Quasiconformal harmonic mappings and close-to-convex domains , Filomat (2010), 63–68. OMPLETELY MONOTONE SEQUENCES AND HARMONIC MAPPINGS 15
11. R. K¨ustner,
On the order of starlikeness of the shifted Gauss hypergeometric function , J. Math. Anal.Appl. (2007), 1363–1385.12. J. L. Lewis,
Convexity of a certain series , J. London Math. Soc. (1983), 435–446.13. J.-G. Liu and R. L. Pego, On generating functions of Hausdorff moment sequences , Trans. Amer.Math. Soc. (2016), 8499–8518.14. B.-Y. Long and M. Dorff,
Linear combinations of a class of harmonic univalent mappings , Filomat (2018), 3111–3121.15. D. Partyka, K. Sakan, and J.-F. Zhu, Quasiconformal harmonic mappings with the convex holomor-phic part , Ann. Acad. Sci. Fenn. Math. (2018), 401–418.16. A. Peyerimhoff, On the modulus of power series of a certain type , J. London Math. Soc. (1965),260–261.17. M. R. Reza and G. Zhang, Hausdorff moment sequences induced by rational functions , ComplexAnalysis and Operator Theory (2019), 4117–4142.18. O. Roth, St. Ruscheweyh, and L. Salinas, A note on generating functions for Hausdorff momentsequences , Proc. Amer. Math. Soc. (2008), 3171–3176.19. St. Ruscheweyh,
Convolutions in Geometric Function Theory , S´eminaire de Math´ematiquesSup´erieures, vol. 83, Les Presses de l’Universit´e de Montr´eal, Montr´eal, 1982.20. St. Ruscheweyh,
Some properties of prestarlike and universally prestarlike functions , J. Anal. (2007), 247–254.21. St. Ruscheweyh, L. Salinas, and T. Sugawa, Completely monotone sequences and universally prestar-like functions , Israel J. Math. (2009), 285–304.22. J. A. Shohat and J. D. Tamarkin,
The Problem of Moment , AMS, 1943.23. T. Sugawa and L.-M. Wang,
Geometric properties of the shifted hypergeometric functions , ComplexAnalysis and Operator Theory (2017), 1879–1893.24. T. Sugawa and L.-M. Wang, Spirallikeness of shifted hypergeometric functions , Ann. Acad. Sci. Fenn.Math. (2017), 963–977.25. Y. Sun, A. Rasila, and Y.-P. Jiang, Linear combinations of harmonic quasiconformal mappings convexin one direction , Kodai Math. J. (2016), 366–377.26. H. S. Wall, Analytic Theory of Continued Fractions , D. Van Nostrand Company, Inc., New York, N.Y., 1948.27. K.-J. Wirths, ¨Uber totalmonotone Zahlenfolgen , Arch. Math. (Basel) (1975), 508–517. School of Mathematical Sciences, Anhui University, Hefei 230601, China
Email address : [email protected] Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, Japan
Email address : [email protected] School of Mathematical Sciences, Anhui University, Hefei 230601, China
Email address ::