Table Of ContentBOUNDS FOR RADII OF STARLIKENESS OF SOME q-BESSEL FUNCTIONS
I˙BRAHI˙MAKTAS¸ ANDA´RPA´DBARICZ⋆
Abstract. InthispapertheradiiofstarlikenessoftheJacksonandHahn-Extonq-Besselfunctionsare
considered andforeach of them three differentnormalization areapplied. ByapplyingEuler-Rayleigh
inequalities for the first positive zeros of these functions tight lower and upper bounds for the radii of
7 starlikeness of these functions are obtained. The Laguerre-P´olya class of real entirefunctions plays an
1 importantroleinthisstudy. Inparticular,weobtainsomenewboundsforthefirstpositivezeroofthe
0 derivativeoftheclassicalBesselfunctionofthefirstkind.
2
1. Introduction
n
a Let D be the open disk z C : z < r with radius r > 0. Let denote the class of analytic
r
J functions f : Dr C, which{sat∈isfy th|e|norma}lization conditions f(0) =Af′(0) 1 = 0. By we mean
8 the classoffuncti→ons belongingto whichareunivalentinD andlet ∗ be the−subclassof Sconsisting
1 of functions which are starlike withArespect to origin in D . Trhe analytSic characterizationofSthis class of
r
] functions is
A zf′(z)
∗ = f :Re >0 for all z D .
r
C S ∈S f(z) ∈
(cid:26) (cid:18) (cid:19) (cid:27)
. The real number
h zf′(z)
at r∗(f)=sup r >0 Re f(z) >0for all z ∈Dr
m (cid:26) (cid:12) (cid:18) (cid:19) (cid:27)
is called the radius of starlikeness of the fun(cid:12)ction f. Note that r∗(f) is the largest radius such that
(cid:12)
[ the image region f(Dr∗(f)) is a starlike doma(cid:12)in with respect to the origin. For more information about
starlike functions we refer to Duren’s book [17] and to the references therein.
1
v Now, consider the Jackson and Hahn-Exton q-Bessel functions which are defined as follow:
29 J(2)(z;q)= (qν+1;q)∞ (−1)n z2 2n+ν qn(n+ν)
0 ν (q;q) (q;q) (qν+1;q)
5 ∞ nX≥0 n (cid:0) (cid:1) n
0 and
1. J(3)(z;q)= (qν+1;q)∞ (−1)nz2n+ν q12n(n+1),
0 ν (q;q)∞ (q;q)n(qν+1;q)n
n≥0
7 X
where z C,ν > 1,q (0,1) and
1
∈ − ∈
: n
v (a;q) =1,(a;q) = 1 aqk−1 ,(a,q) = 1 aqk−1 .
i 0 n − ∞ −
X
k=1 k≥1
Y(cid:0) (cid:1) Y(cid:0) (cid:1)
r It is known that the Jackson and Hahn-Exton q-Bessel functions are q-extensions of the classical Bessel
a
functionofthefirstkindJ . Clearly,forfixedzwehaveJ(2)((1 z)q;q) J (z)andJ(3)((1 z)q;q)
ν ν ν ν
− → − →
J (2z) as q 1. The readerscanfind comprehensiveinformationon the Besselfunction ofthe firstkind
ν
ր
in Watson’s treatise [25] and properties of Jackson and Hahn-Exton q-Bessel functions can be found in
[18, 19, 21, 22] and in the references therein. The geometric properties of some special functions (like
Bessel,StruveandLommelfunctionsofthe firstkind)andtheirzerosinconnectionwiththesegeometric
properties were intensively studied by many authors (see [1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16,
20, 23, 26]). Also, the radii of starlikeness and convexity of some q-Bessel functions were investigated in
[8]. In some of the above mentioned papers it was shown that the radii of univalence, starlikeness and
File: bounds.tex, printed: 2017-01-19, 1.23
2010 Mathematics Subject Classification. 30C45,30C15, 33C10.
Keywords andphrases. Starlikefunctions;radiusofstarlikeness;Mittag-Lefflerexpansions;q-Besselfunctions;zerosof
q-Besselfunctions;Laguerre-P´olyaclassofentirefunctions.
⋆TheresearchofA´.BariczwassupportedbyaresearchgrantoftheBabe¸s-BolyaiUniversityforyoungresearcherswith
projectnumberGTC-31777.
1
2 I˙.AKTAS¸ANDA´.BARICZ
convexity are actually solutions of some transcendental equations. In addition, it was shown that the
obtained radii satisfy some inequalities. On the other hand, it was proved that the radii of univalence
andstarlikenessofsome normalizedBesselandStruve functions ofthe firstkindcoincide. Mostofabove
papers benefited from some properties of the positive zeros of Bessel, Struve and Lommel functions of
the first kind. Also, the Laguerre-Po´lyaclass of realentire functions, which consist of uniform limits
LP
of realpolynomials whose zerosare allreal, wasused intensively (for more details on the Laguerre-Po´lya
class of entire functions we refer to [8] and to the references therein). Motivated by the earlier works,
in this study our aim is to obtain some lower and upper bounds for the radii of starlikeness of some
normalized q-Bessel functions. The results presented in this paper complement the results of [8] about
the radii of starlikeness and extend the known results from [2] on classical Bessel functions of the first
kind to q-Bessel functions. As in [8] we consider three different normalized forms of Jackson and Hahn-
Extonq-Besselfunctions which are analyticin the unit disk of the complex plane. Becausethe functions
J(2)(.;q) and J(3)(.;q) do not belong to , first we consider the following six normalized forms as in [8].
ν ν
A
For ν > 1,
−
1
f(2)(z;q)= 2νc (q)J(2)(z;q) ν ,ν =0
ν ν ν 6
(cid:16) (cid:17)
g(2)(z;q)=2νc (q)z1−νJ(2)(z;q),
ν ν ν
h(ν2)(z;q)=2νcν(q)z1−ν2Jν(2)(√z;q),
1
f(3)(z;q)= c (q)J(3)(z;q) ν ,ν =0
ν ν ν 6
(cid:16) (cid:17)
g(3)(z;q)=c (q)z1−νJ(3)(z;q),
ν ν ν
h(ν3)(z;q)=cν(q)z1−ν2Jν(3)(√z;q),
wherec (q)=(q;q) (qν+1;q) . Inthisway,alloftheabovefunctionsbelongtotheclass . Ofcourse
ν ∞ ∞
A
there exist infinitely many other normalizationfor both Jacksonand Hahn-Extonq-Besselfunctions, the
(cid:14)
main motivation to consider these six functions is the fact that their limiting cases for Bessel functions
appear in literature, see for example [16] and the references therein.
2. Bounds for the radii of starlikeness of some normalized q-Bessel functions
In this section our aim is to present some tight lower and upper bounds for the radii of starlikeness
of the above mentioned six normalized forms of the Jackson and Hahn-Exton q-Bessel functions. In
particular, we obtain some known and new bounds for the first positive zero of the first derivative of
the classical Bessel function J . We note that the implicit representation of the radii of starlikeness
ν
considered in this section were found in [8].
Theorem 1. Let ν >0. Then the radius of starlikeness r∗ f(2)(z;q) of the function
ν
(cid:16) (cid:17)
1
z f(2)(z;q)= 2νc (q)J(2)(z;q) ν
7→ ν ν ν
(cid:16) (cid:17)
is the smallest positive root of the equation r dJ(2)(r;q)/dr =0 and satisfies the following inequality
ν
·
4ν qν+1 1 (q 1) 2 4ν(ν+2) qν+1 1 qν+2 1 (q2 1)
− − < r∗ f(2)(z;q) < − − − .
qν+1(ν+2) ν qν+1((ν+2)2(1+q qν+2) 2ν(ν+4)q2+(ν2+4ν 4)qν+3)
(cid:0) (cid:1) (cid:16) (cid:16) (cid:17)(cid:17) −(cid:0) − (cid:1)(cid:0) (cid:1) −
It is worth to mention that by multiplying by (1 q)−2 both sides of the above inequality and taking
−
the limit as q 1 for ν >0 we obtain
ր
(2.1) 4ν(ν+1) < j′ 2 < 4ν(ν+1)(ν+2)2,
ν+2 ν,1 ν2+8ν+8
which were obtained earlier by Ismail and M(cid:0)uldo(cid:1)on [20]. Here j′ is the first positive zero of J′.
ν,1 ν
Theorem 2. Let ν > 1. Then the radius of starlikeness r∗ g(2)(z;q) of the function
ν
−
(cid:16) (cid:17)
z g(2)(z;q)=2νc (q)z1−νJ(2)(z;q)
7→ ν ν ν
BOUNDS FOR RADII OF STARLIKENESS OF SOME q-BESSEL FUNCTIONS 3
is the smallest positive root of the equation r dJ(2)(r;q)/dr (ν 1)J(2)(r;q) = 0 and satisfies the
ν ν
· − −
following inequalities
(qν+1 1)(q 1) 3(qν+1 1)(qν+2 1)(1 q2)
2 − − <r∗ g(2)(z;q) <2 − − −
s 3qν+1 ν sqν+1(9q 9qν+2+qν+3 10q2+9)
(cid:16) (cid:17) − −
and
(qν+1 1)2(1 qν+2)(q 1)2(q+1) 2 3(1 qν+1)(qν+3 1)(q 1)
24 − − − <r∗ g(2)(z;q) < − − − q ∗(ν),
sq2ν+2(9q 9qν+2+qν+3 10q2+9) ν 3s (q+1)qν+1 1
− − (cid:16) (cid:17)
where
(q3+2q2+2q+1)(9q 9qν+2+qν+3 10q2+9)
q ∗(ν)= − − ,
1
a (q)+b (q)
ν ν
a (q)= 9qν+2 12qν+3 21qν+4+3qν+5+6qν+6+qν+7+9q2ν+5+3q2ν+6,
ν
− − −
b (q)=3q2ν+7+q2ν+8+18q+3q2 6q3 15q4+7q6+9.
ν
− −
Note that by multiplying by (1 q)−1 both sides of the above inequalities and taking the limit as
−
q 1 for ν > 1 we obtain the first two inequalities of [2, Theorem 1], namely:
ր −
ν+1 3(ν+1)(ν+2)
(2.2) 2 <r⋆(ϕ )<2
ν
3 4ν+13
r r
and
(ν+1)2(ν+2) (ν+1)(ν+3)(4ν+13)
(2.3) 24 <r⋆(ϕ )<2 ,
r 4ν+13 ν s 2(4ν2+26ν+49)
where r⋆(ϕ ) stands for the radii of starlikeness of the normalized Bessel function
ν
z ϕ (z)=2νΓ(ν+1)z1−νJ (z).
ν ν
7→
Theorem 3. Let ν > 1. Then the radius of starlikeness r∗ h(2)(z;q) of the function
ν
−
z 7→h(ν2)(z;q)=2νcν(q)z1−ν2J(cid:16)ν(2)(√z;q)(cid:17)
is the smallest positive root of the equation r dJ(2)(r;q)/dr (ν 2)J(2)(r;q) = 0 and satisfies the
ν ν
· − −
following inequalities
2(1 q)(1 qν+1) 4(qν+1 1)(qν+2 1)(1 q2)
− − <r∗ h(2)(z;q) < − − −
qν+1 ν qν+1(2q 2qν+2+qν+3 3q2+2)
(cid:16) (cid:17) − −
and
8(qν+1 1)2(1 qν+2)(q 1)2(q+1) 4(qν+1 1)(qν+3 1)(q3 1)
− − − <r∗ h(2)(z;q) < − − − q ∗(ν),
s q2ν+2(2q 2qν+2+qν+3 3q2+2) ν qν+1 2
− − (cid:16) (cid:17)
where
2q 2qν+2+qν+3 3q2+2
q ∗(ν)= − − ,
2
c (q)+d (q)
ν ν
c (q)=4qν+2+3qν+3+7qν+4 6qν+5 5qν+6+3qν+7 4q2ν+5+q2ν+6,
ν
− − −
d (q)=q2ν+7 q2ν+8 8q+q2+5q3+9q4 6q6 4.
ν
− − − −
Here we wouldlike to emphasize that by multiplying by (1 q)−2 both sides of the above inequalities
−
and taking the limit as q 1 for ν > 1 we obtain the first two inequalities of [2, Theorem 2], namely:
ր −
8(ν+1)(ν+2)
(2.4) 2(ν+1)<r⋆(φ )<
ν
ν+5
and
4(ν+1)√ν+2 4(ν+1)(ν+3)(ν+5)
(2.5) <r⋆(φ )< ,
√ν+5 ν ν2+8ν+23
where r⋆(φ ) stands for the radii of starlikeness of the normalized Bessel function
ν
z φν(z)=2νΓ(ν+1)z1−ν2Jν(√z).
7→
4 I˙.AKTAS¸ANDA´.BARICZ
Theorem 4. Let ν >0. Then the radius of starlikeness r∗ f(3)(z;q) of the function
ν
(cid:16) 1(cid:17)
z f(3)(z;q)= c (q)J(3)(z;q) ν
7→ ν ν ν
(cid:16) (cid:17)
is the smallest positive root of the equation r dJ(3)(r;q)/dr =0 and satisfies the following inequality
ν
·
ν(1 q)(1 qν+1) 2 ν(ν+2)(1 q2)(1 qν+1)(1 qν+2)
− − < r∗ f(3)(z;q) < − − − .
q(ν+2) ν q((1+q)(1 qν+2)(ν+2)2 2ν(ν+4)q(1 qν+1))
(cid:16) (cid:16) (cid:17)(cid:17) − − −
Multiplying by (1 q)−2 both sides of the above inequality and taking the limit as q 1 for ν > 0
− ր
we obtain the following inequality
(2.6) ν(ν+1) < j′ 2 < ν(ν +1)(ν+2)2.
ν+2 ν,1 ν2+8ν+8
(cid:0) (cid:1)
Theorem 5. Let ν > 1. Then the radius of starlikeness r∗ g(3)(z;q) of the function
ν
−
(cid:16) (cid:17)
z g(3)(z;q)=c (q)z1−νJ(3)(z;q)
7→ ν ν ν
is the smallest positive root of the equation r dJ(3)(r;q)/dr (ν 1)J(3)(r;q) = 0 and satisfies the
ν ν
· − −
following inequalities
(1 q)(1 qν+1) 3(qν+1 1)(qν+2 1)(q2 1)
− − <r∗ g(3)(z;q) < − − −
s 3q ν s q(9qν+3 qν+2+q 9)
(cid:16) (cid:17) − −
and
(qν+1 1)2(q 1)2(qν+2 1)(q+1) (q 1)(qν+1 1)(qν+3 1)
4 − − − <r∗ g(3)(z;q) < − − − q ∗(ν),
s q2(9qν+3 qν+2+q 9) ν s 3q(q+1) 3
− − (cid:16) (cid:17)
where
(9qν+3 qν+2+q 9)(q3+2q2+2q+1)
q ∗(ν)= − − ,
3
s (q)+r (q)
ν ν
s (q)=6qν+2 12qν+3 20qν+4 12qν+5+6qν+6+q2ν+5,
ν
− − −
r (q)=3q2ν+6+3q2ν+7+9q2ν+8+q3+3q2+3q+9.
ν
Multiplying by (1 q)−1 both sides ofthe aboveinequalities andtaking the limit as q 1 forν > 1
− ր −
we obtain the following inequalities
ν+1 3(ν+1)(ν+2)
(2.7) <r⋆(ϕ )<
ν
3 4ν+13
r r
and
(ν+1)2(ν+2) (ν+1)(ν+3)(4ν+13)
(2.8) 4 <r⋆(ϕ )< .
r 4ν+13 ν s 2(4ν2+26ν+49)
Theorem 6. Let ν > 1. Then the radius of starlikeness r∗ h(3)(z;q) of the function
ν
−
z 7→h(ν3)(z;q)=cν(q)z1−ν2J(cid:16)ν(3)(√z;q)(cid:17)
is the smallest positive root of the equation r dJ(3)(r;q)/dr (ν 2)J(3)(r;q) = 0 and satisfies the
ν ν
· − −
following inequalities
(1 q)(1 qν+1) (q2 1)(qν+1 1)(qν+2 1)
− − <r∗ h(3)(z;q) < − − −
2q ν q(q qν+2+2qν+3 2)
(cid:16) (cid:17) − −
and
(q 1)(qν+1 1) 2(qν+2 1)(q+1) (qν+1 1)(qν+3 1)(1 q)
− − − <r∗ h(3)(z;q) < − − − q ∗(ν),
2q sq qν+2+2qν+3 2 ν q(q+1) 4
− − (cid:16) (cid:17)
where
(q qν+2+2qν+3 2)(q3+2q2+2q+1)
q ∗(ν)= − − .
4 q+q2 q3 4 5qν+2+3qν+3+10qν+4+3qν+5 5qν+6 q2ν+5+q2ν+6+q2ν+7 4q2ν+8
− − − − − −
BOUNDS FOR RADII OF STARLIKENESS OF SOME q-BESSEL FUNCTIONS 5
Multiplying by (1 q)−2 both sides ofthe aboveinequalities andtaking the limit as q 1 forν > 1
− ր −
we obtain the following inequalities
ν+1 2(ν+1)(ν+2)
(2.9) <r⋆(φ )<
ν
2 ν+5
and
ν+2 (ν+1)(ν+3)(ν+5)
(2.10) (ν+1) <r⋆(φ )< .
ν+5 ν ν2+8ν+23
r
It is important to mention that by making a comparison among of above obtained inequalities we
have that the left-hand side of (2.6) is weaker than the known result of Ismail and Muldoon, stated as
the left-hand side of (2.1). However, the right-hand side of (2.6) improves the known result of Ismail
and Muldoon, stated as the right-hand side of (2.1). On the other hand, the left-hand side of (2.7) is
weaker than the left-hand side of (2.2), while the right-hand side of (2.7) improves the right-hand side
of (2.2) Also, the left-hand side of (2.8) is weaker than the left-hand side of (2.3), while the right-hand
side of (2.8) improves the right-hand side of (2.3). Finally, the left-hand side of (2.9) is weaker than the
left-hand side of (2.4). However,the right-handside of (2.9) improvesthe right-handside of (2.4). Also,
the left-hand side of (2.10) is weaker than the left-hand side of (2.5), while the right-hand side of (2.10)
improves the right-hand side of (2.5).
3. Proofs of main results
In this section we are going to present the proofs of our main results.
Proof of Theorem 1. Inview of[8, Theorem1]weknowthatthe radiusofstarlikenessofthe function
f(2)(;q) is the smallest positive root of the equation r dJ(2)(r;q)/dr = 0. Also from [8, eq. (2.2)] we
ν ν
· ·
have that
2νc (q)dJ(2)(z;q) z2
ν ν
(3.1) (z,q)= = 1 .
Bν νzν−1 dz − j′ (q) 2!
n≥1 ν,n
Y
In addition, by using the definition of the Jackson q-Bessel function(cid:0) we obt(cid:1)ain
2νc (q)dJ(2)(z;q) ( 1)n(2n+ν)qn(n+ν)
(3.2) (z,q)= ν ν = − z2n.
Bν νzν−1 dz ν22n(q;q) (qν+1;q)
n n
n≥0
X
Now, taking the logarithmic derivative of (3.1) we get
′(z,q)
(3.3) Bν = 2 ∆ z2k+1, z <j′ (q),
(z,q) − k+1 | | ν,1
ν
B k≥0
X
where ∆ = j′ (q) −2k. On the other hand, by considering (3.2) we obtain that
k n≥1 ν,n
P (cid:0) (cid:1) ′(z,q)
(3.4) Bν = ǫ z2n+1 µ z2n,
n n
(z,q)
ν ,
B n≥0 n≥0
X X
where
( 1)n+1(2n+ν+2)(n+1)q(n+1)(n+ν+1) ( 1)n(2n+ν)qn(n+ν)
ǫ = − and µ = − .
n ν22n+1(q;q) (qν+1;q) n ν22n(q;q) (qν+1;q)
n+1 n+1 n n
By equating (3.3) and (3.4) and making the Cauchy product we obtain the following Euler -Rayleigh
sums ∆ = j′ (q) −2k in terms of ν and q for k 1,2 . Namely,
k n≥1 ν,n ∈{ }
P (cid:0) (cid:1) (ν+2)qν+1
∆ = ,
1 4ν(1 q)(1 qν+1)
− −
and
q2ν+2 (ν+2)2q 2ν(ν+4)q2 (ν+2)2qν+2+(ν2+4ν 4)qν+3+(ν+2)2
∆ = − − − .
2 16ν2(1 qν+1)(qν+2 1)(1 q)2(1+q)
(cid:0) − − − (cid:1)
By considering the above Euler-Rayleighsums and using the Euler-Rayleighinequalities
∆−k1 < j′ (q) 2 < ∆k
k ν,1 ∆
k+1
(cid:0) (cid:1)
6 I˙.AKTAS¸ANDA´.BARICZ
for ν > 1 we get the following inequality
−
4ν qν+1 1 (q 1) 2 4ν(ν+2) qν+1 1 qν+2 1 (q2 1)
− − < r∗ f(2)(z;q) < − − − .
qν+1(ν+2) ν qν+1((ν+2)2(1+q qν+2) 2ν(ν+4)q2+(ν2+4ν 4)qν+3)
(cid:0) (cid:1) (cid:16) (cid:16) (cid:17)(cid:17) −(cid:0) − (cid:1)(cid:0) (cid:1) −
Itispossibletohavemoretighterboundsfortheradiusofstarlikenessofthenormalizedq-Besselfunction
(2)
f for other values of k, but it would be quite complicated, this is why we restricted ourselves to the
ν
first Euler-Rayleigh inequality. (cid:3)
Proof of Theorem 2. Inview of[8, Theorem1]weknowthatthe radiusofstarlikenessofthe function
(2) (2) (2)
g (;q) is the smallest positive rootof the equationr dJ (r;q)/dr (ν 1)J (r;q)=0. Now, recall
ν ν ν
that·the zeros j (q),n N, of the Jackson q-Bessel f·unction are all−real−and simple, according to [18,
ν,n
∈
Theorem4.2]. Then,thefunctiong(2)(;q)belongstotheLaguerre-Po´lyaclass ofrealentirefunctions.
ν
· LP
Since is closed under differentiation the function z dg(2)(z;q)/dz belongs also to the class .
ν
LP 7→ LP
Hence the function z dg(2)(z;q)/dz has only real zeros. Also its growth order ρ is 0, that is
ν
7→
nlogn
ρ= lim =0,
n→∞2nlog2+log(q;q)n+log(qν+1;q)n log(2n+1) n(n+ν)logq
− −
since as n we have (q;q) (q;q) < and (qν+1;q) (qν+1;q) < . Now, by applying
n ∞ n ∞
→ ∞ → ∞ → ∞
Hadamard’s Theorem [24, p. 26] we obtain
dg(2)(z;q) z2
ν
φ (z;q)= = 1 ,
ν dz − α2 (q)
n≥1(cid:18) ν,n (cid:19)
Y
whereα (q)isthenthzeroofthefunctionφ (;q). Now,vialogarithmicderivationofφ (;q)weobtain
ν,n ν ν
· ·
φ′(z;q)
(3.5) ν = 2 δ z2k+1, z <α (q),
k+1 ν,1
φ (z;q) − | |
ν
k≥0
X
where δ = (α (q))−2k. Also, by using the infinite sum representation of φ we get
k n≥1 ν,n ν
P φ′(z;q)
(3.6) ν = u z2n+1 v z2n,
n n
φ (z;q)
ν ,
n≥0 n≥0
X X
where
( 1)n+1(2n+2)(2n+3)q(n+1)(n+ν+1) ( 1)n(2n+1)qn(n+ν)
u = − and v = − .
n 22n+2(q;q) (qν+1;q) n 22n(q;q) (qν+1;q)
n+1 n+1 n n
By comparing (3.5) and (3.6) and matching all terms with the same degree we have the following Euler-
Rayleigh sums δ = α−2k(q) in terms of ν and q. That is,
k n≥1 ν,n
P 3qν+1
δ = ,
1 4(qν+1 1)(q 1)
− −
q2ν+2(9q 9qν+2+qν+3 10q2+9)
δ = − −
2 −16(qν+1 1)2(qν+2 1)(q 1)2(q+1)
− − −
and
3q3(ν+1)(9q 9qν+2+qν+3 10q2+9)
δ = − − .
3 64(qν+1 1)3(qν+2 1)(qν+3 1)(q 1)3q∗(ν)
− − − − 1
Now, by considering these Euler-Rayleigh sums in the known Euler-Rayleighinequalities
δ−k1 <α2 (q)< δk
k ν,1 δ
k+1
for ν > 1 and k 1,2 we obtain the inequalities of this theorem. (cid:3)
− ∈{ }
Proof of Theorem 3. Byputting α=0inpartcof[8,Theorem1]wehavethatthe radiusofstarlike-
nessr∗ h(2)(z;q) ofthefunctionh(2)(;q)isthesmallestpositiverootoftheequationr dJ(2)(r;q)/dr
ν ν ν
· · −
(ν 2)J(cid:16)(2)(r;q)=(cid:17)0. It is known that the function
ν
−
( 1)n(n+1)znqn(n+ν)
(3.7) Ψ (z;q)=dh(2)(z;q)/dz = −
ν ν 22n(q;q) (qν+1;q)
n n
n≥0
X
BOUNDS FOR RADII OF STARLIKENESS OF SOME q-BESSEL FUNCTIONS 7
is an entire function of order ρ=0, because
nlogn
lim =0
n→∞log(q;q)n+log(qν+1;q)n+2nlog2 n(n+ν)logq log(n+1)
− −
since as n we have (q;q) (q;q) < and (qν+1;q) (qν+1;q) < . Moreover, we know
n ∞ n ∞
→ ∞ → ∞ → ∞
that the zeros β (q),n N, of the function z dh(2)(z;q)/dz are real when ν > 1, according to [8,
ν,n ν
∈ 7→ −
Lemma 6]. Now, by applying Hadamard’s Theorem [24, p. 26] we get
z
(3.8) Ψ (z;q)=dh(2)(z;q)/dz = 1 .
ν ν − β2 (q)
n≥1(cid:18) ν,n (cid:19)
Y
From (3.7) we have
Ψ′(z;q)
(3.9) ν = t zn m zn,
n n
Ψ (z;q)
ν ,
n≥0 n≥0
X X
where
( 1)n+1(n+2)(n+1)q(n+1)(n+ν+1) ( 1)n(n+1)qn(n+ν
t = − and m = − .
n 22n+2(q;q) (qν+1;q) n 22n(q;q) (qν+1;q)
n+1 n+1 n n
Also, taking the derivative of (3.8) logarithmically we get
Ψ′(z;q)
(3.10) ν = Θ zk, z <β2 (q),
Ψ (z;q) − k+1 | | ν,1
ν
k≥0
X
where Θ = β−2k(q). Now, it is possible to state the Euler-Rayleigh sums Θ = β−2k(q) in
k n≥1 ν,n k n≥1 ν,n
terms of ν and q. By comparisonof the coefficients of (3.9) and (3.10) we obtain
P P
qν+1 q2ν+2(2q 2qν+2+qν+3 3q2+2)
Θ = and Θ = − − .
1 2(1 q)(1 qν+1) 2 8(qν+1 1)(1 qν+2)(q 1)2(q+1)
− − − − −
By considering the Euler-Rayleighinequalities
Θ−k1 <r∗ h(2)(z;q) < Θk
k ν Θ
k+1
(cid:16) (cid:17)
for ν > 1 and k 1,2 we get the following inequalities
− ∈{ }
2(1 q)(1 qν+1) 4(qν+1 1)(qν+2 1)(1 q2)
− − <r∗ h(2)(z;q) < − − −
qν+1 ν qν+1(2q 2qν+2+qν+3 3q2+2)
(cid:16) (cid:17) − −
and
8(qν+1 1)2(1 qν+2)(q 1)2(q+1) 4(qν+1 1)(qν+3 1)(q3 1)
− − − <r∗ h(2)(z;q) < − − − q ∗(ν).
s q2ν+2(2q 2qν+2+qν+3 3q2+2) ν qν+1 2
− − (cid:16) (cid:17)
(cid:3)
Proof of Theorem 4. Bytakingα=0inpartaof[8,Theorem1]wehavethattheradiusofstarlikeness
r∗ f(3)(z;q) of the function f(3)(;q) is the smallest positive rootof the equationr dJ(3)(r;q)/dr =0.
ν ν ν
· ·
Als(cid:16)o, from th(cid:17)e equation (2.2) in [8] we have that
c (q) dJ(3)(z;q) z2
ν ν
(3.11) (z,q)= = 1 .
Kν νzν−1 dz − l′ (q) 2!
n≥1 ν,n
Y
On the other hand, by using the infinite sum representation of t(cid:0)he Hah(cid:1)n-Exton q-Bessel function we
obtain
(3.12) (z,q)= cν(q) dJν(3)(z;q) = (−1)n(2n+ν)q12n(n+1)z2n.
Kν νzν−1 dz ν(q;q) (qν+1;q)
n n
n≥0
X
Taking the logarithmic derivative of (3.11) we have
′(z,q)
(3.13) Kν = 2 ξ z2k+1, z <l′ (q),
(z,q) − k+1 | | ν,1
ν
K k≥0
X
8 I˙.AKTAS¸ANDA´.BARICZ
where ξ = l′ (q) −2k. By using the expression (3.12) we write
k n≥ ν,n
P (cid:0) (cid:1) ′(z,q)
(3.14) Kν = s z2n+1 p z2n,
n n
(z,q)
ν ,
K n≥0 n≥0
X X
where
( 1)n+12(n+1)(2n+ν+2)q12(n+1)(n+2) ( 1)n(2n+ν)q12n(n+1)
s = − and p = − .
n ν(q;q) (qν+1;q) n ν(q;q) (qν+1;q)
n+1 n+1 n n
Now, it is possible to express the Euler-Rayleigh sums ξ = l′ (q) −2k in terms of ν and q. By
k n≥ ν,n
matching the coefficients of the equalities (3.13) and (3.14) we get
P (cid:0) (cid:1)
q(ν+2) (1+q)(1 qν+2)(ν+2)2 2ν(ν+4)q(1 qν+1)
ξ = and ξ =q2 − − − .
1 ν(1 q)(1 qν+1) 2 ν2(1 q)2(1+q)(1 qν+1)2(1 qν+2)
− − (cid:18) − − − (cid:19)
By considering the above Euler-Rayleighsums in the Euler-Rayleigh inequalities
ξ−k1 <r∗ f(3)(z;q) < ξk
k ν ξ
k+1
(cid:16) (cid:17)
for ν >0 and k =1 we have the following inequality
ν(1 q)(1 qν+1) 2 ν(ν +2)(1 q2)(1 qν+1)(1 qν+2)
− − <r∗ f(3)(z;q) < − − − .
q(ν+2) ν q((1+q)(1 qν+2)(ν+2)2 2ν(ν+4)q(1 qν+1))
(cid:16) (cid:17) − − − (cid:3)
Proof of Theorem 5. By virtue of partb of [8, Theorem1] the radius of starlikenessr∗ g(3)(z;q) of
ν
the functiong(3)(;q)is the smallestpositive rootofthe equationr dJ(3)(r;q)/dr (ν 1)(cid:16)J(3)(r;q)=(cid:17) 0.
ν ν ν
Now, recall that ·the zeros j (q),n N, of the Hahn-Exton q-Be·ssel function a−re al−l real and simple,
ν,n
∈
according to [18, Theorem 4.2]. Then, the function g(3)(;q) belongs to the Laguerre-Po´lya class of
ν
· LP
real entire functions. Since is closed under differentiation the function
LP
(3.15) z Φ (z;q)= dgν(3)(z;q) = (−1)n(2n+1)q21n(n+1)z2n
7→ ν dz (q;q) (qν+1;q)
n n
n≥0
X
belongs also to the class . Hence the function z dg(3)(z;q)/dz has only real zeros. Also its growth
ν
LP 7→
order ρ is 0, that is
nlogn
ρ= lim =0,
n→∞log(q;q)n+log(qν+1;q)n− 21n(n+1)logq−log(2n+1)
since as n we have (q;q) (q;q) < and (qν+1;q) (qν+1;q) < . Due to Hadamard’s
n ∞ n ∞
→ ∞ → ∞ → ∞
Theorem [24, p. 26] we have
z2
(3.16) Φ (z;q)= 1 .
ν − γ2 (q)
n≥(cid:18) ν,n (cid:19)
Y
If we consider the equality (3.15) we can write that
Φ′(z;q)
(3.17) ν = k z2n+1 l z2n,
n n
Φ (z;q)
ν ,
n≥0 n≥0
X X
where
( 1)n+1(2n+2)(2n+3)q21(n+1)(n+2) ( 1)n(2n+1)q12n(n+1)
k = − and l = − .
n (q;q) (qν+1;q) n (q;q) (qν+1;q)
n+1 n+1 n n
In addition, by using the logarithmic derivative of both sides of (3.16) we have
Φ′(z;q)
(3.18) ν = 2 S z2k+1, z <γ (q),
k+1 ν,1
Φ (z;q) − | |
ν
k≥0
X
where S = γ−2k(q). Now, it is possible to express the Euler-Rayleigh sums S = γ−2k(q)
k n≥1 ν,n k n≥1 ν,n
in terms of ν and q. By matching the coefficients of the equalities (3.17) and (3.18) we get
P P
3q q2(9qν+3 qν+2+q 9)
S = ,S = − −
1 (1 q)(1 qν+1) 2 (qν+1 1)2(qν+2 1)(q 1)2(q+1)
− − − − −
BOUNDS FOR RADII OF STARLIKENESS OF SOME q-BESSEL FUNCTIONS 9
and
3q3(9qν+3 qν+2+q 9)
S = − − .
3 (qν+1 1)3(qν+2 1)(qν+3 1)(q 1)3q ∗(ν)
3
− − − −
By considering the above Euler-Rayleighsums in the Euler-Rayleigh inequalities
S−k1 < r∗ g(3)(z;q) 2 < Sk
k ν S
k+1
(cid:16) (cid:16) (cid:17)(cid:17)
for ν > 1 and k 1,2 we have the following inequalities
− ∈{ }
(1 q)(1 qν+1) 3(qν+1 1)(qν+2 1)(q2 1)
− − <r∗ g(3)(z;q) < − − −
s 3q ν s q(9qν+3 qν+2+q 9)
(cid:16) (cid:17) − −
and
(qν+1 1)2(q 1)2(qν+2 1)(q+1) (q 1)(qν+1 1)(qν+3 1)
4 − − − <r∗ g(3)(z;q) < − − − q ∗(ν).
s q2(9qν+3 qν+2+q 9) ν s 3q(q+1) 3
− − (cid:16) (cid:17)
(cid:3)
Proof of Theorem 6. Thanks to part c of [8, Theorem 1] we know that the radius of starlikeness
r∗ h(3)(z;q) of the function h(3)(;q) is the smallest positive root of the equation r dJ(3)(r;q)/dr
ν ν ν
· · −
(ν(cid:16) 2)J(3)(r(cid:17);q) = 0. On the other hand, with the help of the infinite sum representation of the Hahn-
ν
−
Exton q-Bessel function the function z dh(3)(z;q)/dz can be written as an infinite sum as follow:
ν
7→
(3.19) Υ (z;q)= dh(ν3)(z;q) = (−1)n(n+1)q21n(n+1)zn.
ν dz (q;q) (qν+1;q)
n n
n≥0
X
Also its growth order ρ is 0, that is
nlogn
ρ= lim =0,
n→∞log(q;q)n+log(qν+1;q)n− 12n(n+1)logq−log(n+1)
since as n we have (q;q) (q;q) < and (qν+1;q) (qν+1;q) < . In addition, it is
n ∞ n ∞
known that→th∞e zeros γ (q) of t→he function Υ∞(z;q) are real for→ν > 1 and n ∞N, according to [8,
ν,n ν
− ∈
Lemma 6]. By applying Hadamard’s Theorem [24, p. 26] we have
z
(3.20) Υ (z;q)= 1 .
ν − γ2 (q)
n≥1(cid:18) ν,n (cid:19)
Y
Now, using the equality (3.19) we have
Υ′(z;q)
(3.21) ν = x zn y zn,
n n
Υ (z;q)
ν ,
n≥0 n≥0
X X
where
( 1)n+1(n+1)(n+2)q21(n+1)(n+2) ( 1)n(n+1)q21n(n+1)
x = − and y = − .
n (q;q) (qν+1;q) n (q;q) (qν+1;q)
n+1 n+1 n n
Then, taking the logarithmic derivative of (3.20) we get
Υ′(z;q)
(3.22) ν = σ zk, z <γ2 (q),
Υ (z;q) − k+1 | | ν,1
ν
k≥0
X
where σ = γ−2k(q). By comparing allcoefficients of (3.21) and (3.22) it is possible to expressthe
k n≥1 ν,n
Euler-Rayleighsums σ = γ−2k(q) in terms of ν and q. Namely,
P k n≥1 ν,n
P 6q3 2q2(q qν+2+2qν+3 2)
σ = ,σ = − −
1 (q 1)(1 q2)(1 qν+1)(1 qν+2) 2 (qν+1 1)2(qν+2 1)(q 1)2(q+1)
− − − − − − −
and
2q3(q qν+2+2qν+3 2)
σ = − − .
3 (qν+1 1)3(qν+2 1)(1 qν+3)(q 1)3q∗(ν)
− − − − 4
Now, applying the Euler-Rayleighinequalities
σ−k1 <r∗ h(3)(z;q) < σk
k ν σ
k+1
(cid:16) (cid:17)
10 I˙.AKTAS¸ANDA´.BARICZ
for k 1,2 we get the following inequalities
∈{ }
(q 1)(1 q2)(1 qν+1)(1 qν+2) (q2 1)(qν+1 1)(qν+2 1)
− − − − <r∗ h(3)(z;q) < − − −
6q3 ν q(q qν+2+2qν+3 2)
(cid:16) (cid:17) − −
and
(q 1)(qν+1 1) 2(qν+2 1)(q+1) (qν+1 1)(qν+3 1)(1 q)
− − − <r∗ h(3)(z;q) < − − − q ∗(ν).
2q sq qν+2+2qν+3 2 ν q(q+1) 4
− − (cid:16) (cid:17)
(cid:3)
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Departmentof MathematicalEngineering, Faculty of Engineering and Natural Sciences, Gu¨mu¨¸shane Uni-
versity, Gu¨mu¨¸shane,Turkey
E-mail address: [email protected]
Departmentof Economics, Babe¸s-BolyaiUniversity, Cluj-Napoca,Romania
Institute of Applied Mathematics,O´buda University,Budapest, Hungary
E-mail address: [email protected]
