See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/263163764 Radii of starlikeness of some special functions Article  in  Proceedings of the American Mathematical Society · June 2014 DOI: 10.1090/proc/13120 · Source: arXiv CITATIONS 13 READS 105 4 authors: Some of the authors of this publication are also working on these related projects: Geometric properties of generalized Bessel functions View project The Nyman-Beurling and Báez-Duarte criteria for zeros of Dirichlet L-functions View project Árpád Baricz Babeş-Bolyai University 157 PUBLICATIONS   1,723 CITATIONS    SEE PROFILE Dimitar K. Dimitrov São Paulo State University 100 PUBLICATIONS   640 CITATIONS    SEE PROFILE Halit Orhan Ataturk University 111 PUBLICATIONS   743 CITATIONS    SEE PROFILE Nihat Yağmur 23 PUBLICATIONS   268 CITATIONS    SEE PROFILE All content following this page was uploaded by Dimitar K. Dimitrov on 25 June 2014. 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C A ] 1 4 Ju n 20 14 RADII OF STARLIKENESS OF SOME SPECIAL FUNCTIONS ÁRPÁD BARICZ, DIMITAR K. DIMITROV, HALIT ORHAN, AND NIHAT YAGMUR Abstract. Geometric properties of the classical Lommel and Struve functions, both of the first kind, are studied. For each of them, there different normalizations are applied in such a way that the resulting functions are analytic in the unit disc of the complex plane. For each of the six functions we determine the radius of starlikeness precisely. 1. Introduction and statement of the main results Let Dr be the open disk {z ∈ C : |z| < r} , where r > 0, and set D = D1. By A we mean the class of analytic functions f : Dr → C which satisfy the usual normalization conditions f(0) = f ′(0) − 1 = 0. Denote by S the class of functions belonging to A which are univalent in Dr and let S∗(α) be the subclass of S consisting of functions which are starlike of order α in Dr, where 0 ≤ α < 1. The analytic characterization of this class of functions is S∗(α) = { f ∈ S : ℜ ( zf ′(z) f(z) ) > α for all z ∈ Dr } , and we adopt the convention S∗ = S∗(0). The real number r∗α(f) = sup { r > 0 : ℜ ( zf ′(z) f(z) ) > α for all z ∈ Dr } , is called the radius of starlikeness of order α of the function f. Note that r∗(f) = r∗0(f) is the largest radius such that the image region f(Dr∗(f)) is a starlike domain with respect to the origin. We consider two classical special functions, the Lommel function of the first kind sµ,ν and the Struve function of the first kind Hν . They are explicitly defined in terms of the hypergeometric function 1F2 by (1.1) sµ,ν(z) = zµ+1 (µ− ν + 1)(µ+ ν + 1) 1F2 ( 1; µ− ν + 3 2 , µ+ ν + 3 2 ;−z 2 4 ) , 1 2 (−µ± ν − 3) 6∈ N, and (1.2) Hν(z) = ( z 2 )ν+1 √ π 4 Γ ( ν + 3 2 ) 1F2 ( 1; 3 2 , ν + 3 2 ;−z 2 4 ) , −ν − 3 2 6∈ N. A common feature of these functions is that they are solutions of inhomogeneous Bessel differential equations [Wa]. Indeed, the Lommel function of the first kind sµ,ν is a solution of z2w′′(z) + zw′(z) + (z2 − ν2)w(z) = zµ+1 while the Struve function Hν obeys z2w′′(z) + zw′(z) + (z2 − ν2)w(z) = 4 ( z 2 )ν+1 √ πΓ ( ν + 1 2 ) . We refer to Watson’s treatise [Wa] for comprehensive information about these functions and recall some more recent contributions. In 1972 Steinig [St] examined the sign of sµ,ν(z) for real µ, ν and positive z. He showed, among other things, that for µ < 1 2 the function sµ,ν has infinitely many changes of sign on (0,∞). In 2012 Koumandos and Lamprecht [KL] obtained sharp estimates for the location of the zeros of sµ− 1 2 , 1 2 when µ ∈ (0, 1). The Turán type inequalities for sµ− 1 2 , 1 2 were established in [BK] while those for the Struve function were proved in [BPS]. 2010 Mathematics Subject Classification. 30C45, 30C15. Key words and phrases. Lommel functions of the first kind; Struve functions; univalent, starlike functions; radius of starlikeness; zeros of Lommel functions of the first kind; zeros of Struve functions; trigonometric integrals. The research of Á. Baricz is supported by the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, under Grant PN-II-RU-TE-2012-3-0190. The research of D. K. Dimitrov is supported by the Brazilian foundations CNPq under Grant 307183/2013–0 and FAPESP under Grants 2009/13832–9. 1 http://de.arxiv.org/abs/1406.3732v1 2 Á. BARICZ, D. K. DIMITROV, H. ORHAN, AND N. YAGMUR Geometric properties of sµ− 1 2 , 1 2 and of the Struve function were obtained in [BS] and in [OY, YO], respectively. Motivated by those results we study the problem of starlikeness of certain analytic functions related to the classical special functions under discussion. Since neither sµ,ν , nor Hν belongs to A, first we perform some natural normalizations. We define three functions originating from sµ,ν : fµ,ν(z) = ((µ− ν + 1)(µ+ ν + 1)sµ,ν(z)) 1 µ+1 , gµ,ν(z) = (µ− ν + 1)(µ+ ν + 1)z−µsµ,ν(z) and hµ,ν(z) = (µ− ν + 1)(µ+ ν + 1)z 1−µ 2 sµ,ν( √ z). Similarly, we associate with Hν the functions uν(z) = (√ π2νΓ ( ν + 3 2 ) Hν(z) ) 1 ν+1 , vν(z) = √ π2νz−νΓ ( ν + 3 2 ) Hν(z) and wν(z) = √ π2νz 1−ν 2 Γ ( ν + 3 2 ) Hν( √ z). Clearly the functions fµ,ν , gµ,ν , hµ,ν , uν , vν and wν belong to the class A. The main results in the present note concern the exact values of the radii of starlikeness for these six function, for some ranges of the parameters. Let us set fµ(z) = fµ− 1 2 , 1 2 (z), gµ(z) = gµ− 1 2 , 1 2 (z) and hµ(z) = hµ− 1 2 , 1 2 (z). The first principal result we establish reads as follows: Theorem 1. Let µ ∈ (−1, 1), µ 6= 0. The following statements hold: a) If 0 ≤ α < 1 and µ ∈ ( − 1 2 , 0 ) , then r∗α(fµ) = xµ,α, where xµ,α is the smallest positive root of the equation z s′ µ− 1 2 , 1 2 (z)− α ( µ+ 1 2 ) sµ− 1 2 , 1 2 (z) = 0. Moreover, if 0 ≤ α < 1 and µ ∈ ( −1,− 1 2 ) , then r∗α(fµ) = qµ,α, where qµ,α is the unique positive root of the equation izs′ µ− 1 2 , 1 2 (iz)− α ( µ+ 1 2 ) sµ− 1 2 , 1 2 (iz) = 0. b) If 0 ≤ α < 1, then r∗α(gµ) = yµ,α, where yµ,α is the smallest positive root of the equation z s′ µ− 1 2 , 1 2 (z)− ( µ+ α− 1 2 ) sµ− 1 2 , 1 2 (z) = 0. c) If 0 ≤ α < 1, then r∗α(hµ) = tµ,α, where tµ,α is the smallest positive root of the equation zs′ µ− 1 2 , 1 2 (z)− ( µ+ 2α− 3 2 ) sµ− 1 2 , 1 2 (z) = 0. The corresponding result about the radii of starlikeness of the functions, related to Struve’s one, is: Theorem 2. Let |ν| < 1 2 . The following assertions are true: a) If 0 ≤ α < 1, then r∗α(uν) = δν,α, where δν,α is the smallest positive root of the equation zH′ ν(z)− α(ν + 1)Hν(z) = 0. b) If 0 ≤ α < 1, then r∗(vν) = ρν,α, where ρν,α is the smallest positive root of the equation zH′ ν(z)− (α+ ν)Hν(z) = 0. c) If 0 ≤ α < 1, then r∗α(wν) = σν,α, where σν,α is the smallest positive root of the equation zH′ ν(z)− (2α+ ν − 1)Hν(z) = 0. It is worth mentioning that the starlikeness of hµ, when µ ∈ (−1, 1), µ 6= 0, as well as of wν , under the restriction |ν| ≤ 1 2 , were established in [BS], and it was proved there that all the derivatives of these functions are close-to-convex in D. RADII OF STARLIKENESS OF SOME SPECIAL FUNCTIONS 3 2. Preliminaries 2.1. The Hadamard’s factorization. The following preliminary result is the content of Lemmas 1 and 2 in [BK]. Lemma 1. Let ϕk(z) = 1F2 ( 1; µ− k + 2 2 , µ− k + 3 2 ;−z 2 4 ) where z ∈ C, µ ∈ R and k ∈ {0, 1, . . . } such that µ− k is not in {0,−1, . . .}. Then, ϕk is an entire function of order ρ = 1 and of exponential type τ = 1. Consequently, the Hadamard’s factorization of ϕk is of the form (2.1) ϕk(z) = ∏ n≥1 ( 1− z2 z2µ,k,n ) , where ±zµ,k,1, ±zµ,k,2, . . . are all zeros of the function ϕk and the infinite product is absolutely convergent. Moreover, for z, µ and k as above, we have (µ− k + 1)ϕk+1(z) = (µ− k + 1)ϕk(z) + zϕ′ k(z), √ zsµ−k− 1 2 , 1 2 (z) = zµ−k+1 (µ− k)(µ− k + 1) ϕk(z). 2.2. Quotients of power series. We will also need the following result (see [BK, PV]): Lemma 2. Consider the power series f(x) = ∑ n≥0 anx n and g(x) = ∑ n≥0 bnx n, where an ∈ R and bn > 0 for all n ≥ 0. Suppose that both series converge on (−r, r), for some r > 0. If the sequence {an/bn}n≥0 is increasing (decreasing), then the function x 7→ f(x)/g(x) is increasing (decreasing) too on (0, r). The result remains true for the power series f(x) = ∑ n≥0 anx 2n and g(x) = ∑ n≥0 bnx 2n. 2.3. Zeros of polynomials and entire functions and the Laguerre-Pólya class. In this subsection we provide the necessary information about polynomials and entire functions with real zeros. An algebraic polynomial is called hyperbolic if all its zeros are real. The simple statement that two real polynomials p and q posses real and interlacing zeros if and only if any linear combinations of p and q is a hyperbolic polynomial is sometimes called Obrechkoff’s theorem. We formulate the following specific statement that we shall need. Lemma 3. Let p(x) = 1−a1x+a2x2−a3x3+ · · ·+(−1)nanx n = (1−x/x1) · · · (1−x/xn) be a hyperbolic polynomial with positive zeros 0 < x1 ≤ x2 ≤ · · · ≤ xn, and normalized by p(0) = 1. Then, for any constant C, the polynomial q(x) = Cp(x) − x p′(x) is hyperbolic. Moreover, the smallest zero η1 belongs to the interval (0, x1) if and only if C < 0. The proof is straightforward; it suffices to apply Rolle’s theorem and then count the sign changes of the linear combination at the zeros of p. We refer to [BDR, DMR] for further results on monotonicity and asymptotics of zeros of linear combinations of hyperbolic polynomials. A real entire function ψ belongs to the Laguerre-Pólya class LP if it can be represented in the form ψ(x) = cxme−ax2+βx ∏ k≥1 ( 1 + x xk ) e − x xk , with c, β, xk ∈ R, a ≥ 0, m ∈ N∪{0}, ∑ x−2 k <∞. Similarly, φ is said to be of type I in the Laguerre-Pólya class, written ϕ ∈ LPI, if φ(x) or φ(−x) can be represented as φ(x) = cxmeσx ∏ k≥1 ( 1 + x xk ) , with c ∈ R, σ ≥ 0, m ∈ N ∪ {0}, xk > 0, ∑ 1/xk < ∞. The class LP is the complement of the space of hyperbolic polynomials in the topology induced by the uniform convergence on the compact sets of 4 Á. BARICZ, D. K. DIMITROV, H. ORHAN, AND N. YAGMUR the complex plane while LPI is the complement of the hyperbolic polynomials whose zeros posses a preassigned constant sign. Given an entire function ϕ with the Maclaurin expansion ϕ(x) = ∑ k≥0 γk xk k! , its Jensen polynomials are defined by gn(ϕ;x) = gn(x) = n ∑ j=0 ( n j ) γjx j . Jensen proved the following relation in [Jen]: Theorem A. The function ϕ belongs to LP (LPI, respectively) if and only if all the polynomials gn(ϕ;x), n = 1, 2, . . ., are hyperbolic (hyperbolic with zeros of equal sign). Moreover, the sequence gn(ϕ; z/n) converges locally uniformly to ϕ(z). Further information about the Laguerre-Pólya class can be found in [Obr, RS] while [DC] contains references and additional facts about the Jensen polynomials in general and also about those related to the Bessel function. A special emphasis has been given on the question of characterizing the kernels whose Fourier transform belongs to LP (see [DR]). The following is a typical result of this nature, due to Pólya [Po]. Theorem B. Suppose that the function K is positive, strictly increasing and continuous on [0, 1) and integrable there. Then the entire functions U(z) = ∫ 1 0 K(t) sin(zt)dt and V (z) = ∫ 1 0 K(t) cos(zt)dt have only real and simple zeros and their zeros interlace. In other words, the latter result states that both the sine and the cosine transforms of a kernel are in the Laguerre-Pólya class provided the kernel is compactly supported and increasing in the support. Theorem 3. Let µ ∈ (−1, 1), µ 6= 0, and c be a constant such that c < µ + 1 2 . Then the functions z 7→ zs′ µ− 1 2 , 1 2 (z)− csµ− 1 2 , 1 2 (z) can be represented in the form (2.2) µ(µ+ 1) ( z s′ µ− 1 2 , 1 2 (z)− c sµ− 1 2 , 1 2 (z) ) = zµ+ 1 2ψµ(z), where ψµ is an even entire function and ψµ ∈ LP. Moreover, the smallest positive zero of ψµ does not exceed the first positive zero of sµ− 1 2 , 1 2 . Similarly, if |ν| < 1 2 and d is a constant satisfying d < ν + 1, then (2.3) √ π 2 Γ ( ν + 3 2 ) ( zH′ ν(z)− dHν(z) ) = (z 2 )ν+1 φν(z), where φν is an entire function in the Laguerre-Pólya class and the smallest positive zero of φν does not exceed the first positive zero of Hν . Proof. First suppose that µ ∈ (0, 1). Since, by (1.1), µ(µ+ 1)sµ− 1 2 , 1 2 (z) = ∑ k≥0 (−1)kz2k+µ+ 1 2 22k ( µ+2 2 ) k ( µ+3 2 ) k , then µ(µ+ 1)z s′ µ− 1 2 , 1 2 (z) = ∑ k≥0 (−1)k ( 2k + µ+ 1 2 ) z2k+µ+ 1 2 22k ( µ+2 2 ) k ( µ+3 2 ) k . Therefore, (2.2) holds with ψµ(z) = ∑ k≥0 2k + µ+ 1 2 − c ( µ+2 2 ) k ( µ+3 2 ) k ( −z 2 4 )k . On the other hand, by Lemma 1, µ(µ+ 1)sµ− 1 2 , 1 2 (z) = zµ+ 1 2ϕ0(z), RADII OF STARLIKENESS OF SOME SPECIAL FUNCTIONS 5 and, by [BK, Lemma 3], we have (2.4) zϕ0(z) = µ(µ+ 1) ∫ 1 0 (1− t)µ−1 sin(zt)dt, for µ > 0. Therefore ϕ0 has the Maclaurin expansion ϕ0(z) = ∑ k≥0 1 ( µ+2 2 ) k ( µ+3 2 ) k ( −z 2 4 )k . Moreover, (2.4) and Theorem B imply that ϕ0 ∈ LP for µ ∈ (0, 1), so that the function ϕ̃0(z) := ϕ0(2 √ z), ϕ̃0(ζ) = ∑ k≥0 1 ( µ+2 2 ) k ( µ+3 2 ) k (−ζ)k , belongs to LPI. Then it follows form Theorem A that its Jensen polynomials gn(ϕ̃0; ζ) = n ∑ k=0 ( n k ) k! ( µ+2 2 ) k ( µ+3 2 ) k (−ζ)k are all hyperbolic. However, observe that the Jensen polynomials of ψ̃µ(z) := ψµ(2 √ z) are simply −1 2 gn(ψ̃µ; ζ) = −1 2 ( µ+ 1 2 − c ) gn(ϕ̃0; ζ)− ζ g′n(ϕ̃0; ζ). Lemma 3 implies that all zeros of gn(ψ̃µ; ζ) are real and positive and that the smallest one precedes the first zero of gn(ϕ̃0; ζ). In view of Theorem A, the latter conclusion immediately yields that ψ̃µ ∈ LPI and that its first zero precedes the one of ϕ̃0. Finally, the first statement of the theorem for µ ∈ (0, 1) follows after we go back from ψ̃µ and ϕ̃0 to ψµ and ϕ0 by setting ζ = − z2 4 . Now we prove (2.2) for the case when µ ∈ (−1, 0). Observe that for µ ∈ (0, 1) the function [BK, Lemma 3] ϕ1(z) = ∑ k≥0 1 ( µ+1 2 ) k ( µ+2 2 ) k ( −z 2 4 )k = µ ∫ 1 0 (1− t)µ−1 cos(zt)dt belongs also to Laguerre-Pólya class LP , and hence the Jensen polynomials of ϕ̃1(z) := ϕ1(2 √ z) are hyperbolic. Straightforward calculations show that the Jensen polynomials of ψ̃µ−1(z) := ψµ−1(2 √ z) are −1 2 gn(ψ̃µ−1; ζ) = −1 2 ( µ− 1 2 − c ) gn(ϕ̃1; ζ)− ζ g′n(ϕ̃1; ζ). Lemma 3 implies that for µ ∈ (0, 1) all zeros of gn(ψ̃µ−1; ζ) are real and positive and that the smallest one precedes the first zero of gn(ϕ̃1; ζ). This fact, together with Theorem A, yields that ψ̃µ−1 ∈ LPI and that its first zero precedes the one of ϕ̃1. Consequently, the first statement of the theorem for µ ∈ (−1, 0) follows after we go back from ψ̃µ−1 and ϕ̃1 to ψµ−1 and ϕ1 by setting ζ = − z2 4 and substituting µ by µ+ 1. In order to prove the corresponding statement for (2.3), we recall first that the hypergeometric repre- sentation (1.2) of the Struve function is equivalent to √ π 2 Γ ( ν + 3 2 ) Hν(z) = ∑ k≥0 (−1)k ( 3 2 ) k ( ν + 3 2 ) k (z 2 )2k+ν+1 , which immediately yields φν(z) = ∑ k≥0 2k + ν + 1− d ( 3 2 ) k ( ν + 3 2 ) k ( −z 2 4 )k . On the other hand, the integral representation Hν(z) = 2 ( z 2 )ν √ πΓ ( ν + 1 2 ) ∫ 1 0 (1− t2)ν− 1 2 sin(zt)dt, which holds for ν > − 1 2 , and Theorem B imply that the even entire function Hν(z) = ∑ k≥0 1 ( 3 2 ) k ( ν + 3 2 ) k ( −z 2 4 )k 6 Á. BARICZ, D. K. DIMITROV, H. ORHAN, AND N. YAGMUR belongs to the Laguerre-Pólya class when |ν| < 1 2 . Then the functions H̃ν(z) := Hν(2 √ z), H̃ν(ζ) = ∑ k≥0 1 ( 3 2 ) k ( ν + 3 2 ) k (−ζ)k , is in LPI. Therefore, its Jensen polynomials gn(H̃ν ; ζ) = n ∑ k=0 ( n k ) k! ( 3 2 ) k ( ν + 3 2 ) k (−ζ)k are hyperbolic, with positive zeros. Then, by Lemma 3, the polynomial − 1 2 (ν + 1− d) gn(H̃ν ; ζ) − ζ g′n(H̃ν ; ζ) possesses only real positive zeros. Obviously the latter polynomial coincides with the nth Jensen polynomials of φ̃ν(z) = φν(2 √ z), that is −1 2 gn(φ̃ν ; ζ) = −1 2 (ν + 1− d) gn(H̃ν ; ζ)− ζ g′n(H̃ν ; ζ). Moreover, the smallest zero of gn(φ̃ν ; ζ) precedes the first positive zero of gn(H̃ν ; ζ). This implies that φν ∈ LP and that its first positive zero is smaller that the one of Hν . � 3. Proofs of the main results Proof of Theorem 1. We need to show that for the corresponding values of µ and α the inequalities (3.1) ℜ ( zf ′ µ(z) fµ(z) ) > α, ℜ ( zg′µ(z) gµ(z) ) > α and ℜ ( zh′µ(z) hµ(z) ) > α are valid for z ∈ Dr∗α(fµ), z ∈ Dr∗α(gµ) and z ∈ Dr∗α(hµ) respectively, and each of the above inequalities does not hold in larger disks. It follows from (2.1) that fµ(z) = fµ− 1 2 , 1 2 (z) = ( µ(µ+ 1)sµ− 1 2 , 1 2 (z) ) 1 µ+1 2 = z (ϕ0(z)) 1 µ+1 2 , gµ(z) = gµ− 1 2 , 1 2 (z) = µ(µ+ 1)z−µ+ 1 2 sµ− 1 2 , 1 2 (z) = zϕ0(z), hµ(z) = hµ− 1 2 , 1 2 (z) = µ(µ+ 1)z 3−2µ 4 sµ− 1 2 , 1 2 ( √ z) = zϕ0( √ z), which in turn imply that zf ′ µ(z) fµ(z) = 1 + zϕ′ 0(z) (µ+ 1 2 )ϕ0(z) = 1− 1 µ+ 1 2 ∑ n≥1 2z2 z2µ,0,n − z2 , zg′µ(z) gµ(z) = 1 + zϕ′ 0(z) ϕ0(z) = 1− ∑ n≥1 2z2 z2µ,0,n − z2 , zh′µ(z) hµ(z) = 1 + 1 2 √ zϕ′ 0( √ z) ϕ0( √ z) = 1− ∑ n≥1 z z2µ,0,n − z , respectively. We note that for µ ∈ (0, 1) the function ϕ0 has only real and simple zeros (see [BK]). For µ ∈ (0, 1), and n ∈{1, 2, . . .} let ξµ,n = zµ,0,n be the nth positive zero of ϕ0. We know that (see [KL, Lemma 2.1]) ξµ,n ∈ (nπ, (n + 1)π) for all µ ∈ (0, 1) and n ∈{1, 2, . . . }, which implies that ξµ,n > ξµ,1 > π > 1 for all µ ∈ (0, 1) and n ≥ 2. On the other hand, it is known that [BKS] if z ∈ C and β ∈ R are such that β > |z|, then (3.2) |z| β − |z| ≥ ℜ ( z β − z ) . Then the inequality |z|2 ξ2µ,n − |z|2 ≥ ℜ ( z2 ξ2µ,n − z2 ) , holds get for every µ ∈ (0, 1), n ∈ N and |z| <ξµ,1. Therefore, ℜ ( zf ′ µ(z) fµ(z) ) = 1− 1 µ+ 1 2 ℜ   ∑ n≥1 2z2 ξ2µ,n − z2   ≥ 1− 1 µ+ 1 2 ∑ n≥1 2 |z|2 ξ2µ,n − |z|2 = |z| f ′ µ(|z|) fµ(|z|) , RADII OF STARLIKENESS OF SOME SPECIAL FUNCTIONS 7 ℜ ( zg′µ(z) gµ(z) ) = 1−ℜ   ∑ n≥1 2z2 ξ2µ,n − z2   ≥ 1− ∑ n≥1 2 |z|2 ξ2µ,n − |z|2 = |z| g′µ(|z|) gµ(|z|) and ℜ ( zh′µ(z) hµ(z) ) = 1−ℜ   ∑ n≥1 z ξ2µ,n − z   ≥ 1− ∑ n≥1 |z| ξ2µ,n − |z| = |z|h′µ(|z|) hµ(|z|) , where equalities are attained only when z = |z| = r. The latter inequalities and the minimum principle for harmonic functions imply that the corresponding inequalities in (3.1) hold if and only if |z| < xµ,α, |z| < yµ,α and |z| < tµ,α, respectively, where xµ,α, yµ,α and tµ,α are the smallest positive roots of the equations rf ′ µ(r)/fµ(r) = α, rg′µ(r)/gµ(r) = α, rh′µ(r)/hµ(r) = α. Since their solutions coincide with the zeros of the functions r 7→ rs′ µ− 1 2 , 1 2 (r) − α ( µ+ 1 2 ) sµ− 1 2 , 1 2 (r), r 7→ rs′ µ− 1 2 , 1 2 (r) − ( µ+ α− 1 2 ) sµ− 1 2 , 1 2 (r), r 7→ rs′ µ− 1 2 , 1 2 (r) − ( µ+ 2α− 3 2 ) sµ− 1 2 , 1 2 (r), the result we need follows from Theorem 3. In other words, Theorem 3 show that all the zeros of the above three functions are real and their first positive zeros do not exceed the first positive zero ξµ,1. This guarantees that the above inequalities hold. This completes the proof our theorem when µ ∈ (0, 1). Now we prove that the inequalities in (3.1) also hold for µ ∈ (−1, 0) , except the first one, which is valid for µ ∈ ( − 1 2 , 0 ) . In order to do this, suppose that µ ∈ (0, 1) and adapt the above proof, substituting µ by µ − 1, ϕ0 by the function ϕ1 and taking into account that the nth positive zero of ϕ1, denoted by ζµ,n = zµ,1,n, satisfies (see [BS]) ζµ,n > ζµ,1 > π 2 > 1 for all µ ∈ (0, 1) and n ≥ 2. It is worth mentiontioning that ℜ ( zf ′ µ−1(z) fµ−1(z) ) = 1− 1 µ− 1 2 ℜ   ∑ n≥1 2z2 ζ2µ,n − z2   ≥ 1− 1 µ− 1 2 ∑ n≥1 2 |z|2 ζ2µ,n − |z|2 = |z| f ′ µ−1(|z|) fµ−1(|z|) , remains true for µ ∈ ( 1 2 , 1 ) . In this case we use the minimum principle for harmonic functions to ensure that (3.1) is valid for µ − 1 instead of µ. Thus, using again Theorem 3 and replacing µ by µ + 1, we obtain the statement of the first part for µ ∈ ( − 1 2 , 0 ) . For µ ∈ (−1, 0) the proof of the second and third inequalities in (3.1) go along similar lines. To prove the statement for part a when µ ∈ ( −1,− 1 2 ) we observe that the counterpart of (3.2) is (3.3) Re ( z β − z ) ≥ −|z| β + |z| , and it holds for all z ∈ C and β ∈ R such that β > |z| (see [BKS]). From (3.3), we obtain the inequality ℜ ( z2 ζ2µ,n − z2 ) ≥ −|z|2 ζ2µ,n + |z|2 , which holds for all µ ∈ ( 0, 12 ) , n ∈ N and |z| <ζµ,1 and it implies that ℜ ( zf ′ µ−1(z) fµ−1(z) ) = 1− 1 µ− 1 2 ℜ   ∑ n≥1 2z2 ζ2µ,n − z2   ≥ 1 + 1 µ− 1 2 ∑ n≥1 2 |z|2 ζ2µ,n + |z|2 = i |z| f ′ µ−1(i |z|) fµ−1(i |z|) . In this case equality is attained if z = i |z| = ir. Moreover, the latter inequality implies that ℜ ( zf ′ µ−1(z) fµ−1(z) ) > α if and only if |z| < qµ,α, where qµ,α denotes the smallest positive root of the equation irf ′ µ−1(ir)/fµ−1(ir) = α, which is equivalent to irs′ µ− 3 2 , 1 2 (ir)− α ( µ− 1 2 ) sµ− 3 2 , 1 2 (ir) = 0, for µ ∈ ( 0, 1 2 ) . 8 Á. BARICZ, D. K. DIMITROV, H. ORHAN, AND N. YAGMUR Substituting µ by µ+ 1, we obtain irs′ µ− 1 2 , 1 2 (ir) − α ( µ+ 1 2 ) sµ− 1 2 , 1 2 (ir) = 0, for µ ∈ ( −1,−1 2 ) . It follows from Theorem 3 that the first positive zero of z 7→ izs′ µ− 1 2 , 1 2 (iz) − α ( µ+ 1 2 ) sµ− 1 2 , 1 2 (iz) does not exceed ζµ,1 which guarantees that the above inequalities are valid. All we need to prove is that the above function has actually only one zero in (0,∞). Observe that, according to Lemma 2, the function r 7→ irs′ µ− 1 2 , 1 2 (ir) sµ− 1 2 , 1 2 (ir) is increasing on (0,∞) as a quotient of two power series whose positive coefficients form the increasing “quotient sequence” { 2k + µ+ 1 2 } k≥0 . On the other hand, the above function tends to µ+ 1 2 when r → 0, so that its graph can intersect the horizontal line y = α ( µ+ 1 2 ) > µ + 1 2 only once. This completes the proof of part a of the theorem when µ ∈ (−1, 0). � Proof of Theorem 2. As in the proof of Theorem 1 we need show that, for the corresponding values of ν and α, the inequalities (3.4) ℜ ( zu′ν(z) uν(z) ) > α, ℜ ( zv′ν(z) vν(z) ) > α and ℜ ( zw′ ν(z) wν(z) ) > α are valid for z ∈ Dr∗α(uν), z ∈ Dr∗α(vν) and z ∈ Dr∗α(wν) respectively, and each of the above inequalities does not hold in any larger disk. If |ν| ≤ 1 2 , then (see [BPS, Lemma 1]) the Hadamard factorization of the transcendental entire function Hν , defined by Hν(z) = √ π2νz−ν−1Γ ( ν + 3 2 ) Hν(z), reads as follows Hν(z) = ∏ n≥1 ( 1− z2 h2ν,n ) , which implies that Hν(z) = zν+1 √ π2νΓ ( ν + 3 2 ) ∏ n≥1 ( 1− z2 h2ν,n ) , where hν,n stands for the nth positive zero of the Struve function Hν . We know that (see [BS, Theorem 2]) hν,n > hν,1 > 1 for all |ν| ≤ 1 2 and n ∈ N. If |ν| ≤ 1 2 and |z| < hν,1, then (3.2) imples ℜ ( zu′ν(z) uν(z) ) = 1− 1 ν + 1 ℜ   ∑ n≥1 2z2 h2ν,n − z2   ≥ 1− 1 ν + 1 ∑ n≥1 2 |z|2 h2ν,n − |z|2 = |z|u′ν(|z|) uν(|z|) , ℜ ( zv′ν(z) vν(z) ) = 1−ℜ   ∑ n≥1 2z2 h2ν,n − z2   ≥ 1− ∑ n≥1 2 |z|2 h2ν,n − |z|2 = |z| v′ν(|z|) vν(|z|) and ℜ ( zw′ ν(z) wν(z) ) = 1−ℜ   ∑ n≥1 z h2ν,n − z   ≥ 1− ∑ n≥1 |z| h2ν,n − |z| = |z|w′ ν(|z|) wν(|z|) , where equalities are attained when z = |z| = r. Then minimum principle for harmonic functions implies that the corresponding inequalities in (3.4) hold if and only if |z| < δν,α, |z| < ρν,α and |z| < σν,α, respectively, where δν,α, ρν,α and σν,α are the smallest positive roots of the equations ru′ν(r)/uν(r) = α, rv′ν(r)/vν(r) = α, rw′ ν (r)/wν(r) = α. The solutions of these equations are the zeros of the functions r 7→ rH′ ν(r) − α(ν + 1)Hν(r), r 7→ rH′ ν(r) − (α+ ν)Hν(r), r 7→ rH′ ν(r) − (2α+ ν − 1)Hν(r), which, in view of Theorem 3, have only real zeros and the smallest positive zero of each of them does not exceed the first positive zeros of Hν . � RADII OF STARLIKENESS OF SOME SPECIAL FUNCTIONS 9 References [BK] Á. Baricz, S. Koumandos, Turán type inequalities for some Lommel functions of the first kind, arXiv:1308.6477. [BKS] Á. Baricz, P. A. Kupán, R. Szász, The radius of starlikeness of normalized Bessel functions of the first kind , Proc. Amer. Math. Soc. 142(6) (2014) 2019–2025. [BS] Á. Baricz, R. Szász, Close-to-convexity of some special functions and their derivatives, arXiv:1402.0692. [BPS] Á. Baricz, S. Ponnusamy, S. Singh, Turán type inequalities for Struve functions, arXiv:1401.1430. [BK] M. Biernacki, J. Krzyż, On the monotonity of certain functionals in the theory of analytic function, Ann. Univ. 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Department of Economics, Babeş-Bolyai University, Cluj-Napoca 400591, Romania E-mail address: bariczocsi@yahoo.com Departamento de Matemática Aplicada, IBILCE, Universidade Estadual Paulista UNESP, São José do Rio Preto 15054, Brazil E-mail address: dimitrov@ibilce.unesp.br Department of Mathematics, Ataturk University, Erzurum 25240, Turkey E-mail address: horhan@atauni.edu.tr Department of Mathematics, Erzincan University, Erzincan 24000, Turkey E-mail address: nhtyagmur@gmail.com View publication statsView publication stats https://www.researchgate.net/publication/263163764 1. Introduction and statement of the main results 2. Preliminaries 2.1. The Hadamard's factorization 2.2. Quotients of power series 2.3. Zeros of polynomials and entire functions and the Laguerre-Pólya class 3. Proofs of the main results References