RADII OF STARLIKENESS AND CONVEXITY OF A CROSS-PRODUCT OF BESSEL FUNCTIONS A´RPA´DBARICZANDNIHATYAG˘MUR Abstract. In this paper some geometric properties of the normalized forms of the cross-product and productofBesselandmodifiedBesselfunctionsofthefirstkindarestudied. Forthecross-productand the product threedifferent normalizationareinvestigated andfor each ofthe sixfunctions the radiiof starlikeness and convexity are precisely determined by using their Hadamard factorization. Necessary and sufficient conditions are also given for the parameters such that the six normalized functions are starlikeintheopenunitdisk,howevertheconvexcaseisopenforfurtherresearch. Thecharacterization 6 of entirefunctions fromthe Laguerre-Pólyaclass viahyperbolic polynomials playanimportant rolein 1 thispaper. Moreover,theinterlacingpropertiesofthezerosofthecross-productandproductofBessel 0 functionsandtheirderivativesarealsousefulintheproofofthemainresults. 2 n 1. Introduction and statements of the Main Results a J Let D be the open disk z C: z 0 and D = D. As usual, with we denote the 8 classofarnalyticfunctionsf{:D∈ C|w|hich}satisfytheusualnorm1alizationconditionsf(A0)=f′(0) 1= r A] s0u.bLceltasussodfenotceonbsyisStinthgeocflfausnscotfi→ofunnscwtihoincshbaerleonsgtainrlgikteooAf owrhdiecrhαarienuDniv,awlehnetrein0Dr aαnα for all z ∈Dr , t (cid:26) (cid:18) (cid:19) (cid:27) a and we adopt the convention ∗ = ∗(0). The real number m S S zf′(z) [ r∗(f)=sup r >0 : Re >α for all z D , α f′(z) ∈ r 1 (cid:26) (cid:18) (cid:19) (cid:27) v is called the radius of starlikeness of order α of the function f. It is worth to mention that r∗(f)=r0∗(f) 8 is the largestradius such that the image region f(Dr∗(f)) is a starlike domain with respect to the origin. 9 Also, let (β) be the subclass of consisting of functions which are convex of order β in D , where r 9 K S 0 β <1. The well-known analytic characterizationof this class of functions is 1 ≤ 0 zf′′(z) (β)= f : Re 1+ >β for all z D , 1. K ∈S f′(z) ∈ r (cid:26) (cid:18) (cid:19) (cid:27) 0 and for β =0 it reduces to the class of convex functions. The real number 6 K zf′′(z) 1 rc(f)=sup r >0 : Re 1+ >β for all z D , : β f′(z) ∈ r v (cid:26) (cid:18) (cid:19) (cid:27) i iscalledthe radiusofconvexityoforderβ ofthe function f.Note thatrc(f)=rc(f)isthe largestradius X 0 such that the image region f(D ) is a convex domain in C. rc(f) r a Now, let us consider the Bessel and modified Bessel functions of the first kind (see [14]), which are denoted by J and I , respectively. Moreover, consider the cross-product of Bessel and modified Bessel ν ν functions of the first kind z Φ (z)=J (z)I′(z) J′(z)I (z) 7→ ν ν ν − ν ν and the product function z Π (z)=J (z)I (z). ν ν ν 7→ File: cross_product.tex, printed: 2016-1-11, 1.24 2010 Mathematics Subject Classification. 30C45,33C10. Key words and phrases. Bessel functions ofthefirstkind,cross-productofBesselfunctions, univalent, starlike,convex functions; radius of starlikeness; radius of convexity; zeros of a cross product of Bessel functions; Laguerre-Pólya class of entirefunctions;Laguerreinequality;interlacingpropertyofzeros. ⋆The research of N. Ya˘gmur was supported by Erzincan University Rectorship under “The Scientific and Research ProjectofErzincanUniversity”,projectno. FEN-A-240215-0126. 1 2 A´.BARICZANDN.YAG˘MUR Motivated by their appearance as eigenvalues in the clamped plate problem for the ball, Ashbaugh and Benguria have conjectured that the positive zeros of the function Φ increase with ν on [ 1/2, ), see ν − ∞ [2] for more details. Lorch [13] verified this conjecture and presented some other properties of the zeros of the function Φ . The authors in [1] modified the proof made by Lorch to ν ( 1, ), and deduced ν ∈ − ∞ necessary and sufficient conditions for the close-to-convexity of a normalized form of the functions Φ , ν Π andtheirderivatives. Moreover,veryrecentlyin[6]Bariczetal. wereinterestedonthemonotonicity ν patterns for the cross-product of Bessel and modified Bessel functions, by showing for example that the positive zeros of the cross-product and of the Dini function z (1 ν)J (z)+zJ′(z) are interlacing. 7→ − ν ν In [6] one of the key tools in the proofs of the main results it was the fact that the zeros of of the cross-productare increasingwithν. Motivatedby the aboveresults,in this paper by using amongothers the above monotonicity property of the zeros of the cross-product our aim is to determine the radii of starlikeness and convexity of the normalized forms of the functions Φ and Π . It is worth to mention ν ν that some similar results were obtained recently in the papers [4, 9], in which the authors have studied theradiiofstarlikenessandconvexityofq-BesselfunctionsaswellasforLommelandStruvefunctions of the first kind. We also mention that by using the corresponding recurrence relations for the Bessel and modified Bessel functions of the first kind, then the cross-productcan be rewritten as Φ (z)=J (z)I (z)+J (z)I (z). ν ν+1 ν ν ν+1 Moreover,we know that if z C and ν C such that ν = 1, 2,...then the functions Φ and Π can ν ν ∈ ∈ 6 − − be written as follows (see [17, p. 148] and [1]): ( 1)n z 2ν+4n+1 (1.1) Φ (z)=2 − 2 , ν n!Γ(ν+n+1)Γ(ν+2n+2) n≥0 (cid:0) (cid:1) X and ( 1)n z 2ν+4n (1.2) Π (z)= − 2 . ν n!Γ(ν+n+1)Γ(ν+2n+1) n≥0 (cid:0) (cid:1) X Since neither Φ , nor Π belongs to , first we perform some naturalnormalizations. For ν > 1 we ν ν A − define three functions originating from Φ : ν 1 1 f (z)= 22νΓ(ν+1)Γ(ν+2)Φ (z) 2ν+1 , ν = , ν ν 6 −2 g (cid:2)(z)=22νz−2νΓ(ν+1)Γ(ν+(cid:3) 2)Φ (z) ν ν and hν(z)=22νz−ν2+43Γ(ν+1)Γ(ν+2)Φν(√4z). Similarly, we associate with Π the functions ν 1 u (z)= 22νΓ2(ν+1)Π (z)) 2ν , ν =0, ν ν 6 vν(z)(cid:2)=22νz−2ν+1Γ2(ν+(cid:3)1)Πν(z) and wν(z)=22νz−ν2+1Γ2(ν+1)Πν(√4z). Clearly the functions f , g , h , u , v and w belong to the class . The main results in the present ν ν ν ν ν ν A paperconcerntheexactvaluesoftheradiiofstarlikenessandconvexityforthesesixfunctions. Thepaper is organizedas follows: this section contains the main results of the paper and also some open problems betweenthesecondandthirdsetofmainresults. Section2isdevotedtopresentsomepreliminaryresults, which may be of independent interest, while section 3 contains the proofs of the main results. Thefirstmainresultsweestablishconcernthe radiiofstarlikenessofΦ andΠ ,andreadas follows. ν ν Theorem 1. Let 0 α<1. The following statements hold: ≤ a) If ν > 1, then r∗(f )=x where x is the smallest positive root of the equation −2 α ν ν,α,1 ν,α,1 rΦ′(r) α(2ν+1)Φ (r)=0. ν − ν Moreover, if ν ( 1, 1/2) then r∗(f )=x where x is the unique root of the equation ∈ − − α ν ν,α ν,α i1/2rΦ′(i1/2r) α(2ν+1)Φ (i1/2r)=0. ν − ν b) If ν > 1, then r∗(g )=y where y is the smallest positive root of the equation − α ν ν,α,1 ν,α,1 rΦ′(r) (α+2ν)Φ (r)=0. ν − ν CROSS-PRODUCT OF BESSEL FUNCTIONS 3 c) If ν > 1, then r∗(h )=z where z is the smallest positive root of the equation − α ν ν,α,1 ν,α,1 r1/4Φ′(r1/4) (4α+2ν 3)Φ (r1/4)=0. ν − − ν Theorem 2. Let 0 α<1. The following statements hold: ≤ a) If ν >0, then r∗(u )=δ where δ is the smallest positive root of the equation α ν ν,α,1 ν,α,1 rΠ′(r) 2ανΠ (r)=0. ν − ν Moreover, if ν ( 1,0) then r∗(u )=δ where δ is the unique root of the equation ∈ − α ν ν,α ν,α i1/2rΠ′(i1/2r) 2ανΠ (i1/2r)=0. ν − ν b) If ν > 1, then r∗(v )=ρ where ρ is the smallest positive root of the equation − α ν ν,α,1 ν,α,1 rΠ′(r) (α+2ν 1)Π (r)=0. ν − − ν c) If ν > 1, then r∗(w )=σ where σ is the smallest positive root of the equation − α ν ν,α,1 ν,α,1 r1/4Π′(r1/4) 2(2α+ν 2)Π (r1/4)=0. ν − − ν Our second set of results concerns the radii of convexity. Theorem 3. Suppose that 0 α<1 and let γ and γ′ denote the first positive zeros of Φ and Φ′, ≤ ν,1 ν,1 ν ν respectively. Then the following statements hold: a) If ν > 1/2 then the radius of convexity of order α of the function f is the smallest positive root ν − of the equation rΦ′′(r) 1 rΦ′(r) 1+ ν + 1 ν =α. Φ′(r) 2ν+1 − Φ (r) ν (cid:18) (cid:19) ν b) If ν > 1 then the radius of convexity of order α of the function g is the smallest positive root ν − of the equation (1 2ν)Φ′(r)+rΦ′′(r) 2ν+r − ν ν =α. − 2νΦ (r)+rΦ′(r) − ν ν c) If ν > 1 then the radius of convexity of order α of the function h is the smallest positive root ν − of the equation 3 2ν+r14(4−2ν)Φ′ν(r14)+r14Φ′ν′(r14) =4α. − (3−2ν)Φν(r14)+r14Φ′ν(r14) Moreover, we have the inequalities rc(f )<γ′ <γ , rc(g )<ζ <γ , and rc(h )<ξ <γ α ν ν,1 ν,1 α ν ν,1 ν,1 α ν ν,1 ν,1 where ζ and ξ are the first positive zeros of z zΦ′(z) 2νΦ (z) and z zΦ′(z) (2ν 3)Φ (z). ν,1 ν,1 7→ ν − ν 7→ ν − − ν Theorem 4. Suppose that 0 α < 1. Let j and j′ denote the first positive zeros of J and J′, ≤ ν,1 ν,1 ν ν respectively. Then the following statements hold: a) If ν >0 then the radius of convexity of order α of the function u is the smallest positive root of ν the equation rΠ′′(r) 1 rΠ′(r) 1+ ν + 1 ν =α. Π′(r) 2ν − Π (r) ν (cid:18) (cid:19) ν b) If ν > 1 then the radius of convexity of order α of the function v is the smallest positive root ν − of the equation (2 2ν)Π′(r)+rΠ′′(r) 1 2ν+r − ν ν =α. − (1 2ν)Π (r)+rΠ′(r) − ν ν c) If ν > 1 then the radius of convexity of order α of the function w is the smallest positive root ν − of the equation 4 2ν+r14(5−2ν)Π′ν(r14)+r14Π′ν′(r14) =4α. − (4−2ν)Πν(r14)+r14Π′ν(r14) Moreover, we have rc(u ) 1 and z C then the Hadamard’s factorizations of Φ and Π are ν ν − ∈ z2ν+1 z4 (2.1) Φ (z)= 1 ν 22νΓ(ν+1)Γ(ν+2) − γ4 n≥1(cid:18) ν,n(cid:19) Y and z2ν z4 (2.2) Π (z)= 1 , ν 22νΓ2(ν+1) − j4 n≥1(cid:18) ν,n(cid:19) Y where γ and j are the nth positive zeros of thefunctions Φ and J . Moreover, the zeros γ satisfy ν,n ν,n ν ν ν,n the interlacing inequalities j <γ 1. ν,n ν,n ν,n+1 ν,n ν,n ν+1,n ∈ − 2.2. Quotients of power series. We will also need the following result (see [10, 15]): Lemma 2. Consider the power series f(x) = a xn and g(x) = b xn, where a R and b > 0 n n n n ∈ n≥0 n≥0 X X for all n 0. Suppose that both series converge on ( r,r), for some r >0. If the sequence a /b n n n≥0 ≥ − { } is increasing (decreasing), then the function x f(x)/g(x) is increasing (decreasing) too on (0,r). The 7→ result remains true for the power series f(x)= a x4n and g(x)= b x4n. n n n≥0 n≥0 X X 6 A´.BARICZANDN.YAG˘MUR 2.3. Zeros of hyperbolicpolynomialsand the Laguerre-Po´lya class ofentire functions. Inthis subsection we recall some necessary information about polynomials and entire functions with real zeros. An algebraic polynomial is called hyperbolic if all its zeros are real. We formulate the following specific statement that we shall need, see [3] for more details. Lemma 3. Let p(x)=1 a x+a x2 a x3+ +( 1)na xn =(1 x/x ) (1 x/x )be a hyperbolic 1 2 3 n 1 n − − ··· − − ··· − polynomial with positive zeros 0 < x x x , and normalized by p(0) = 1. Then, for any 1 2 n ≤ ≤ ··· ≤ constant C, the polynomial q(x) = Cp(x) xp′(x) is hyperbolic. Moreover, the smallest zero η belongs 1 − to the interval (0,x ) if and only if C <0. 1 By definition a real entire function ψ belongs to the Laguerre-Po´lyaclass if it can be represented LP in the form ψ(x)=cxme−ax2+βx 1+ x e−xxk, x k≥1(cid:18) k(cid:19) Y with c, β, x R, a 0, m N 0 , x−2 < . Similarly, φ is said to be of type in the k ∈ ≥ ∈ ∪ { } k ∞ I Laguerre-Po´lyaclass, written ϕ , if φ(x) or φ( x) can be represented as ∈LPI P − x φ(x)=cxmeσx 1+ , x k≥1(cid:18) k(cid:19) Y with c R, σ 0, m N 0 , x > 0, 1/x < . The class is the complement of the space k k ∈ ≥ ∈ ∪{ } ∞ LP of hyperbolic polynomials in the topology induced by the uniform convergence on the compact sets of P the complex plane while is the complement of the hyperbolic polynomials whose zeros posses a LPI preassigned constant sign. Given an entire function ϕ with the Maclaurin expansion xn ϕ(x)= τ , n n! n≥0 X its Jensen polynomials are defined by n n P (ϕ;x)=P (x)= τ xj. n n j j j=0(cid:18) (cid:19) X The next result of Jensen [11] is a well-known characterizationof functions belonging to . LP Lemma 4. The function ϕ belongs to ( , respectively) if and only if all the polynomials P (ϕ;x), n LP LPI n = 1,2,..., are hyperbolic (hyperbolic with zeros of equal sign). Moreover, the sequence P (ϕ;z/n) n converges locally uniformly to ϕ(z). The following result is a key tool in the proof of main results. Lemma 5. If ν > 1, then for a<1 the function z (2ν+a)Φ (z) zΦ′(z), and for b<0 the − 7→ ν − ν function z (2ν+b)Π (z) zΠ′(z) can be represented in the form 7→ ν − ν z 2ν+1 Γ(ν+1)Γ(ν+2)[(2ν+a)Φ (z) zΦ′(z)]=2 Ψ (z), ν − ν 2 ν (cid:16) (cid:17) z 2ν Γ2(ν+1)[(2ν+b)Π (z)) zΠ′(z)]= φ (z), ν − ν 2 ν where Ψ and φ are entire functions belonging to the Laguerre-P(cid:16)óly(cid:17)a class . Moreover, the smallest ν ν LP positive zero of Ψ does not exceed γ , while the smallest positive zero of φ is less than j . ν ν,1 ν ν,1 Proof. Suppose that ν > 1. It is clear from Lemma 1 and the infinite product representation of z − 7→ Υ (z) = 22νΓ(ν+1)Γ(ν+2)z−2ν−1Φ (z) that this function belongs to . This implies that the ν ν LP function z Υ (2z1/4) = Υ˜ (z) belongs to . Then it follows from Lemma 4 that its Jensen ν ν 7→ LPI polynomials n n 1 P (Υ˜ ;ζ)= ( ζ)k n ν k (ν+1) (ν+2) − k=0(cid:18) (cid:19) k 2k X are all hyperbolic, where (c) =c(c+1)...(c+k 1) is the well-known Pochhammer symbol. However, k − observe that the Jensen polynomials of z Ψ˜ (z)=Ψ (2z1/4) are simply ν ν 7→ 1 1 P (Ψ˜ ;ζ)= (a 1) P (Υ˜ ;ζ) ζP′(Υ˜ ;ζ). 4 n ν 4 − n ν − n ν CROSS-PRODUCT OF BESSEL FUNCTIONS 7 Lemma 3 implies that all zeros of P (Ψ˜ ;ζ) are real and positive and that the smallest one precedes the n ν first zero of P (Υ˜ ;ζ). In view of Lemma 4, the latter conclusion immediately yields that Ψ˜ n ν ν ∈ LPI and that its first zero is less than γ . Finally, the first part of the statement of the lemma follows after ν,1 we go back from Ψ˜ to Ψ by setting ζ = z4. ν ν 16 Similarly, the function z Ω (z) = 22νΓ2(ν+1)z−2νΠ (z) belongs to the Laguerre-Po´lya class of ν ν 7→ entirefunctions,whichimpliesthatthefunctionz Ω (2z1/4)=Ω (z)belongsto . Thenitfollows ν ν 7→ LPI from Lemma 4 that its Jensen polynomials n n 1 P (Ω ;ζ)= ( ζ)k n ν k (ν+1) (ν+1) − k=0(cid:18) (cid:19) k 2k X are all hyperbolic. However, observe that the Jensen polynomials of z φ˜ (z)=φ (2z1/4) are simply ν ν 7→ 1 b P (φ˜ ;ζ)= P (Ω ;ζ) ζP′(Ω ;ζ). 4 n ν 4 n ν − n ν Lemma 3 implies that all zeros of P (φ˜ ;ζ) are real and positive and that the smallest one precedes the n ν firstzeroofP (Ω ;ζ). Inview ofLemma 4, the latter conclusionimmediately yields thatφ˜ and n ν ν ∈LPI that its first zero precedes j . Thus, the second partof the statement of this lemma follows after we go ν,1 back from φ˜ to φ by setting ζ = z4. (cid:3) ν ν 16 2.4. The Hadamard factorization of the derivatives ofΦ and Π . Thefollowinginfiniteproduct ν ν representations are the well-known Hadamard factorizations for the derivatives of Φ and Π . ν ν Lemma 6. For z C the Hadamard factorizations of Φ′ and Π′ are as follows: if ν > 1/2 then ∈ ν ν − (2ν+1) z 2ν z4 (2.3) Φ′(z)= 2 1 , ν Γ(ν+1)Γ(ν+2) − γ′4 (cid:0) (cid:1) n≥1(cid:18) ν,n(cid:19) Y and if ν >0 then ν z 2ν−1 z4 (2.4) Π′(z)= 2 1 , ν Γ2(ν+1) − t2 (cid:0) (cid:1) n≥1(cid:18) ν,n(cid:19) Y where γ′ ,and t are the nth positive zeros of the functions Φ′ and Π′, respectively. Moreover, if ν,n ν,n ν ν ν > 1/2 then the zeros γ and γ′ interlace; and if ν >0 then the zeros j and t interlace. − ν,n ν,n ν,n ν,n Proof. From (1.1) we have that 1 1 ( 1)n(2ν+4n+1) Γ(ν+1)Γ(ν+2)22νz−2νΦ′(z)= − z4n, 2ν+1 ν 2ν+1 24nn!(ν+1) (ν+2) n≥0 n 2n X and nlogn 1 lim = . n→∞4nlog2+logΓ(n+1)+log(ν+1) +log(ν+2) log(2ν+4n+1) 4 n 2n− Here we used n!=Γ(n+1) and logΓ(bn+c) lim =b, n−→∞ nlogn where b and c are positive constants. In view of the formula of the growth order of entire functions [12, p. 6] we infer that the above entire function is of growthorder ρ= 1, and thus by applying Hadamard’s 4 Theorem [12, p. 26] the infinite product representation of Φ′ can be written indeed as in (2.3). ν Since Υ (z)=22νΓ(ν+1)Γ(ν+2)z−2ν−1Φ (z) belongs to (since the exponential factors in the ν ν infinite product are canceled because of the symmetry of the zLerPos γ , n N, with respect to the ν,n ± ∈ origin), it follows that it satisfies the Laguerre inequality (see [16]) 2 (2.5) Υ(n)(x) Υ(n−1)(x) Υ(n+1)(x) >0, ν − ν ν where ν > 1/2 and z R. O(cid:16)n the oth(cid:17)er ha(cid:16)nd, we hav(cid:17)e(cid:16)that (cid:17) − ∈ Υ′(z)=22νΓ(ν+1)Γ(ν+2)z−2ν−2[zΦ′(z) (2ν+1)Φ (z)], ν ν − ν Υ′′(z)=22νΓ(ν+1)Γ(ν+2)z−2ν−3 z2Φ′′(z) 2(2ν+1)zΦ′(z)+(2ν+1)(2ν+2)Φ (z) , ν ν − ν ν (cid:2) (cid:3) 8 A´.BARICZANDN.YAG˘MUR and thus the Laguerre inequality (2.5) for n=1 is equivalent to 24νΓ2(ν+1)Γ2(ν+2)z−4ν−4 z2(Φ′(z))2 z2Φ (z)Φ′′(z) (2ν+1)(Φ (z)))2 >0. ν − ν ν − ν h i This implies that (2ν+1)(Φ (z))2 (2.6) (Φ′(z))2 Φ (z)Φ′′(z)> ν >0 ν − ν ν z2 for ν > 1/2 and z R, and thus z Φ′(z)/Φ (z) is decreasing on (0, ) γ :n N . Since the zeros γ − of the fun∈ction Φ are rea7→l andν simplνe1, Φ′(z) does not vanis∞h i\n{γν,n, n ∈ N}. Thus, for a fixedνk,n N the function zν Φ′(z)/Φ (z) takes thνe limit when z γ ν,n, and∈the limit when z ∈γ . Moreover, sinc7→e z ν Φ′(z)ν/Φ (z) is decreasing∞on (0, ) ցγ ν,k:−n1 N it results t−h∞at ր ν,k 7→ ν ν ∞ \{ ν,n ∈ } in each interval (γ ,γ ) its restriction intersects the horizontal line only once, and the abscissa of ν,k−1 ν,k this intersection point is exactly γ′ . Consequently, the zeros γ and γ′ interlace. Here we used the ν,k ν,n ν,n convention γ =0. ν,0 By means of (2.2) we have 1 1 ( 1)n(2ν+4n) Γ2(ν+1)22νz−2ν+1Π′(z)= − z4n 2ν ν 2ν 24nn!(ν+1) (ν+1) n≥0 n 2n X of which growth order can be written as nlogn 1 lim = . n→∞4nlog2+logΓ(n+1)+log(ν+1) +log(ν+1) log(2ν+4n) 4 n 2n− By Hadamard’s Theorem [12, p. 26] the infinite product representation of Π′ is exactly as in (2.4). ν Sincez Ω (z)=22νΓ2(ν+1)z−2νΠ (z)belongsto (sincetheexponentialfactorsintheinfinite ν ν product ar7→e canceled because of the symmetry of the zerosLPj , n N, with respect to the origin). On ν,n ± ∈ the other hand, we have that Ω′(z)=22νΓ2(ν+1)z−2ν−1[zΠ′(z) 2νΠ (z)], ν ν − ν Ω′′(z)=22νΓ2(ν+1)z−2ν−2 z2Π′′(z) 4νzΠ′(z)+2ν(2ν+1)Π (z) , ν ν − ν ν and thus the Laguerre inequality (see [16]) for n=1 is equivalent to (cid:2) (cid:3) 24νΓ4(ν+1)z−4ν−2 z2(Π′(z))2 z2Π (z)Π′′(z) 2ν(Π (z)))2 >0. ν − ν ν − ν h i This implies that 2ν(Π (z))2 (Π′(z))2 Π (z)Π′′(z)> ν >0 ν − ν ν z2 for ν >0 and z R, and thus z Π′(z)/Π (z) is decreasing on (0, ) j :n N . Since the zeros j of the ∈function Π are re7→al anνd simpνle, Π′(z) does not vanish∞in\j{ ν,,nn ∈N.}Thus, for a fixed kν,n N the function z ν Π′(z)/Π (z) takes thνe limit when z j ν,n , a∈nd the limit when z ∈ j . Moreover, sin7→ce z ν Π′(zν)/Π (z) is decreasin∞g on (0, )ցj ν,k:−n1 N it results t−h∞at ր ν,k 7→ ν ν ∞ \{ ν,n ∈ } in each interval (j ,j ) its restriction intersects the horizontal line only once, and the abscissa of ν,k−1 ν,k this intersection point is exactly t . Consequently, the zeros j and t interlace. Here we used the ν,k ν,n ν,n convention j =0. (cid:3) ν,0 2.5. The Hadamard factorizations of the derivatives of g , h , v and w . ν ν ν ν Lemma 7. If ν > 1 then the functions g′, h′, v′ and w′ are entire functions of order ρ= 1. Conse- quently, their Hadam−ard factorizations for νz νC aνre of theνform 4 ∈ z4 z g′(z)= 1 , h′(z)= 1 ν − ζ4 ν − ξ4 n≥1(cid:18) ν,n(cid:19) n≥1(cid:18) ν,n(cid:19) Y Y and z4 z v′(z)= 1 , w′(z)= 1 , ν − ϑ4 ν − ω4 n≥1(cid:18) ν,n(cid:19) n≥1(cid:18) ν,n(cid:19) Y Y 1If the zeros γν,n would be not simple, let us suppose that are of multiplicity two for example, then we would have a contradictionwiththeinequality(2.6). CROSS-PRODUCT OF BESSEL FUNCTIONS 9 where ζ and ξ are the nth positive zeros of z zΦ′(z) 2νΦ (z) and z zΦ′(z) (2ν 3)Φ (z) ν,n ν,n 7→ ν − ν 7→ ν − − ν while ϑ and ω are the nth positive zeros of z zΠ′(z) (2ν 1)Π (z) and z zΠ′(z) ν,n ν,n 7→ ν − − ν 7→ ν − (2ν 4)Π (z). ν − Proof. We have that ( 1)n(4n+1)z4n g′(z)=22νΓ(ν+1)Γ(ν+2)z−2ν−1[zΦ′(z) 2νΦ (z)]= − , ν ν − ν n!24n(ν+1) (ν+2) n≥0 n 2n X ( 1)n(n+1)zn h′ν(z)=22ν−2Γ(ν+1)Γ(ν+2)z−ν2−41 z1/4Φ′ν(z1/4)−(2ν−3)Φν(z1/4) = n!24n−(ν+1) (ν+2) , h i nX≥0 n 2n ( 1)n(4n+1)z4n v′(z)=22νΓ2(ν+1)z−2ν[zΠ′(z) (2ν 1)Π (z)]= − , ν ν − − ν n!24n(ν+1) (ν+1) n≥0 n 2n X ( 1)n(n+1)zn wν′(z)=22ν−2Γ2(ν+1)z−ν2 z1/4Π′ν(z1/4)−(2ν−4)Πν(z1/4) = n!24n−(ν+1) (ν+1) , h i nX≥0 n 2n and since nlogn 1 lim = , n→∞4nlog2+logΓ(n+1)+log(ν+1) +log(ν+a) log(bn+1) 4 n 2n− for eacha,b>0, it followsthateachofthe aboveentire functions havegrowthorder1/4andhence their genus is zero. Moreover, we know that the zeros ζ , ξ , ϑ and ω , n N, are real according to ν,n ν,n ν,n ν,n ∈ Lemma 5, and with this the rest of the proof follows by applying Hadamard’s Theorem [12, p. 26]. (cid:3) 3. Proofs of the main results Proof of Theorem 1. First we prove part a for ν > 1/2 and parts b and c for ν > 1. We need to − − show that for the corresponding values of ν and α the inequalities zf′(z) zg′(z) zh′(z) (3.1) Re ν >α, Re ν >α and Re ν >α f (z) g (z) h (z) (cid:18) ν (cid:19) (cid:18) ν (cid:19) (cid:18) ν (cid:19) arevalidforz ∈Drα∗(fν),z ∈Drα∗(gν) andz ∈Drα∗(hν) respectively,andeachoftheaboveinequalitiesdoes not hold in larger disks. It follows from (2.1) that zf′(z) 1 zΦ′(z) 1 4z4 ν = ν =1 , f (z) 2ν+1 Φ (z) − 2ν+1 γ4 z4 ν ν n≥1 ν,n− X zg′(z) zΦ′(z) 4z4 ν = 2ν+ ν =1 , g (z) − Φ (z) − γ4 z4 ν ν n≥1 ν,n− X and zh′(z) 3 ν 1z1/4Φ′(z1/4) z ν = + ν =1 . h (z) 4 − 2 4 Φ (z1/4) − γ4 z ν ν n≥1 ν,n− X On the other hand, it is known that [7] if z C and β R are such that β > z , then ∈ ∈ | | z z (3.2) | | Re . β z ≥ β z −| | (cid:18) − (cid:19) Then the inequality z 4 z4 | | Re , γ4 z 4 ≥ γ4 z4 ν,n−| | (cid:18) ν,n− (cid:19) holds get for every ν > 1, n N and z <γ . Therefore, ν,1 − ∈ | | zf′(z) 1 4z4 1 4 z 4 z f′(z ) (3.3) Re ν =1 Re 1 | | = | | ν | | , (cid:18) fν(z) (cid:19) − 2ν+1 nX≥1γν4,n−z4≥ − 2ν+1nX≥1γν4,n−|z|4 fν(|z|)   zg′(z) 4z4 4 z 4 z g′(z ) (3.4) Re ν =1 Re 1 | | = | | ν | | (cid:18) gν(z) (cid:19) − nX≥1γν4,n−z4≥ −nX≥1γν4,n−|z|4 gν(|z|)   10 A´.BARICZANDN.YAG˘MUR and zh′(z) z z z h′(z ) (3.5) Re ν =1 Re 1 | | = | | ν | | , h (z) −  γ4 z≥ − γ4 z h (z ) (cid:18) ν (cid:19) n≥1 ν,n− n≥1 ν,n−| | ν | | X X   where equalities are attained only when z = z =r. The above inequalities and the minimum principle | | for harmonic functions imply that the corresponding inequalities in (3.1) hold if and only if z 1/2, and parts b and c when ν > 1. − − Now,to provethe statementfor parta when ν 1, 1 we observethatthe counterpartof (3.2) is ∈ − −2 z (cid:0) z(cid:1) (3.6) Re −| | , β z ≥ β+ z (cid:18) − (cid:19) | | and it holds for all z C and β R such that β > z (see [5]). From (3.6), we obtain the inequality ∈ ∈ | | z4 z 4 Re −| | , γ4 z4 ≥ γ4 + z 4 (cid:18) ν,n− (cid:19) ν,n | | which holds for all ν > 1, n N and z <γ and it implies that ν,1 − ∈ | | zf′(z) 1 4z4 1 4 z 4 i1/2 z f′(i1/2 z ) Re ν =1 Re 1+ | | = | | ν | | . (cid:18) fν(z) (cid:19) − 2ν+1 nX≥1γν4,n−z4≥ 2ν+1nX≥1γν4,n+|z|4 fν(i1/2|z|)   In this case equality is attained if z =i1/2 z =i1/2r. Moreover,the above inequality implies that | | zf′(z) Re ν >α f (z) (cid:18) ν (cid:19) if and only if z < x , where x denotes the smallest positive root of i1/2rf′(i1/2r)/f (i1/2r) = α, | | ν,α ν,α ν ν which is equivalent to 1 i1/2rΦ′(i1/2r) α(2ν+1)Φ (i1/2r)=0, for ν 1, . ν − ν ∈ − −2 (cid:18) (cid:19) ItfollowsfromLemma 5thatthe firstpositivezeroofz i1/2zΦ′(i1/2z) α(2ν+1)Φ (i1/2z)does not 7→ ν − ν exceedγ whichguaranteesthatthe aboveinequalitiesarevalid. Allweneedtoproveisthatthe above ν,1 function has actually only one zero in (0, ). Observe that, according to Lemma 2, the function ∞ i1/2rΦ′(i1/2r) (4n+2ν+1) r 4n r 4n r ν = 2 2 7→ Φν(i1/2r) n≥0n!Γ(ν+n+1)Γ(ν(cid:0)+(cid:1)2n+2),n≥0n!Γ(ν+n+(cid:0)1)(cid:1)Γ(ν+2n+2) X X is increasing on (0, ) as a quotient of two power series whose positive coefficients form the increasing ∞ “quotient sequence” 4n+2ν+1 . On the other hand, the above function tends to 2ν+1 when { }n≥0 r 0, so that its graph can intersect the horizontal line y = α(2ν+1) > 2ν+1 only once. This → completes the proof of part a of the theorem when ν 1, 1 . (cid:3) ∈ − −2 Proof of Theorem 2. The proof of Theorem 2 is anal(cid:0)ogous to(cid:1)the proof of Theorem 1. First we prove part a for ν >0 and parts b and c for ν > 1. From (2.2) we have − zu′(z) 1 zΠ′(z) 1 4z4 ν = ν =1 , u (z) 2ν Π (z) − 2ν j4 z4 ν ν n≥1 ν,n− X

Products of Bessel and modified Bessel functions

Similar Products of Bessel and modified Bessel functions