Partial sums of generalized Bessel functions

J ournal of Mathematical I nequalities Volume 8, Number 4 (2014), 863–877

doi:10.7153/jmi-08-65

PARTIAL SUMS OF GENERALIZED BESSEL FUNCTIONS

H ALIT O RHAN AND N IHAT YAGMUR (Communicated by N. Elezović) n Abstract. Let g p,b,c n (z) = z + ∑ bm zm+1 be the sequence of partial sums of generalized m=1

∞

and normalized Bessel functions g p,b,c (z) = z + ∑ bm zm+1 where bm = m=1

(−c/4)m m!(κ )m

and κ :=

p + (b+ 1)/2 = 0,−1,−2,... purpose lower bounds . The of the present paperis to determine g p,b,c (z) g p,b,c (z) (g p,b,c )n (z) (g p,b,c )n (z) , ℜ , ℜ and ℜ . Further we give for ℜ g g p,b,c (z) gp,b,c (z) ( p,b,c )n (z) (g p,b,c )n (z) A[g ](z) (A[g p,b,c ])n (z) lower bounds for ℜ A[g p,b,c] (z) and ℜ , where A[g p,b,c ] is the Alexander A[g p,b,c ](z) ( p,b,c )n transform of g p,b,c .

1. Introduction and preliminary results Let A denote the class of functions f normalized by f (z) = z +

∞

∑ am zm

(1.1)

m=2

which are analytic in the open unit disk U = {z : |z| < 1} and satisfy the usual normalization condition f (0) = f (0) − 1 = 0. Let S denote the subclass of A contains all functions which are univalent in U . Also let S ∗ (α ), C (α ) and K (α ) denote the subclasses of A consisting of functions which are, respectively, starlike, convex and close-to-convex of order α in U (0 α < 1). The Alexander transform A[ f ] : U −→ C of f is defined by A[ f ](z) =

z 0

∞ f (t) am m dt = z + ∑ z . t m=2 m

(1.2)

We consider the following second-order linear homogeneous differential equation (see, for details, [16] and [2]): z2 w (z) + bzw (z) + cz2 − p2 + (1 − b)p w(z) = 0 (1.3) Mathematics subject classification (2010): Primary 30C45; Secondary 33C10. Keywords and phrases: Partial sums, Analytic functions, Generalized Bessel functions, Bessel, modified Bessel and spherical Bessel functions. c , Zagreb Paper JMI-08-65

863

864

H. O RHAN AND N. YAGMUR

where b, c, p ∈ C. A particular solution of the differential equation (1.3), which is denoted by w p,b,c (z), is called the generalized Bessel function of the first kind of order p . In fact we have the following series representation for the function w p,b,c (z) : w p,b,c (z) =

z 2n+p (−c)n (z ∈ C) , ∑ b+1 n=0 n!Γ(p + n + 2 ) 2 ∞

(1.4)

where Γ(z) stands for Euler gamma function. The series in (1.4) permits us to study the Bessel, the modified Bessel and the spherical Bessel functions in a unified manner. Each of these particular cases of the function w p,b,c (z) is worthy of mention here. • For b = c = 1 in (1.4), we obtain the familiar Bessel function J p (z) defined by (see [16] and [2]): z 2n+p ∞ (−1)n (z ∈ C) . (1.5) J p (z) = ∑ n=0 n!Γ(p + n + 1) 2 • For b = −c = 1 in (1.4), we obtain the modified Bessel function I p (z) defined by (see [16] and [2]): I p (z) =

z 2n+p 1 (z ∈ C) . ∑ n=0 n!Γ(p + n + 1) 2 ∞

(1.6)

• For b − 1 = c = 1 in (1.4), we obtain the spherical Bessel function S p (z) defined by (see [16] and [2]): S p (z) =

∞

(−1)n

∑ n!Γ(p + n + 3 )

n=0

z 2n+p

2

2

(z ∈ C) .

(1.7)

We now consider the function g p,b,c (z) defined, in terms of the generalized Bessel function w p,b,c (z), by (see [8]): g p,b,c (z) = 2 p Γ(p +

√ b + 1 1− p )z 2 w p,b,c ( z). 2

(1.8)

By using the Pochhammer (or Appell) symbol, defined in terms of Euler’s gamma functions, by (λ )n = Γ(λ + n)/Γ(λ ) = λ (λ + 1)...(λ + n − 1), we obtain the following series representation for the function g p,b,c (z) given by (1.8): g p,b,c (z) = z +

∞

∑ bm zm+1

(1.9)

m=1 m

and κ := p + (b + 1)/2 = 0, −1, −2, .... where bm = (−c/4) m!(κ )m For further results on this relative g p,b,c (z) of the generalized Bessel function w p,b,c (z), we refer the reader to the recent papers (see, for example, [2, 3, 4, 5, 6, 8, 9]). In this note, we will examine the ratio of a function of the form (1.9) to its sequence n of partial sums g p,b,c n (z) = ∑ bm zm+1 when the coefficients of g p,b,c satisfy some m=0

865

G ENERALIZED B ESSEL FUNCTIONS

g p,b,c (z) (g p,b,c )n (z)

(g p,b,c )n (z)

conditions. We will determine lower bounds for ℜ , ℜ , g p,b,c (z) g p,b,c (z) (g p,b,c )n (z) (A[g p,b,c ])n (z) A[g ](z) ℜ , ℜ , ℜ A[g p,b,c] (z) and ℜ , where A[g p,b,c ](z) gp,b,c (z) ( p,b,c )n (g p,b,c )n (z) A[g p,b,c ] is the Alexander transform of g p,b,c . For various interesting developments concerning partial sums of analytic univalent functions, the reader may be (for examples) refered to the works of Brickman et al. [7], Lin and Owa [10], Orhan and Gunes [11], Owa et.al [12], Sheil-Small [13], Silverman [14], Silvia [15]. L EMMA 1.1. If the parameters b, p ∈ R and c ∈ C are so constrained that κ > |c| 8 ,

then the function g p,b,c : U −→ C

given by (1.9) satisfies the following inequalities: (i) If κ >

(ii) If κ >

|c| 8

then

|c|−4 4

g p,b,c (z) 8κ + |c| (z ∈ U ) , 8κ − |c|

then g

4κ (κ + 1) + (κ + 2)|c| (z ∈ U ) , κ [4(κ + 1) − |c|]

p,b,c (z)

(iii) If κ >

|c| 8

then

A[g p,b,c ](z)

8κ (z ∈ U ) . 8κ − |c|

Proof. (i) By using the well-known triangle inequality: |z1 + z2 | |z1 | + |z2 | and the inequalities m! 2m−1 , we have

(κ )m κ m

(m ∈ N = {1, 2, ...}) ,

∞ ∞ m (−c/4) |c|m m+1 g p,b,c (z) = z + ∑ z 1+ ∑ m m=1 m! (κ )m m=1 m!4 (κ )m m−1 |c| |c| ∞ = 1+ ∑ 8κ 4κ m=1

8κ + |c| |c| , κ> = . 8κ − |c| 8

866


(ii) Suppose that κ > lowing inequality:

|c|−2 4 ,

by using well-known triangle inequality and the fol-

2m!(κ + 1)m−1 (m + 1)(κ + 1)m−1 (m ∈ N) , we get

∞ ∞ m (m + 1)(−c/4) (m + 1)|c|m m z 1+ ∑ p,b,c (z) = 1 + ∑ m m! (κ )m m=1 m=1 m!4 (κ )m

m−1 |c| (m + 1)|c|m−1 |c| ∞ |c| ∞ = 1+ ∑ 2m!4m−1 (κ + 1) 1 + 2κ ∑ 4 (κ + 1) 2κ m=1 m−1 m=1

4κ (κ + 1) + (κ + 2)|c| |c| − 4 = κ> . κ [4(κ + 1) − |c|] 4

g

(iii) In order to prove the part (iii) of Lemma 1.1, we make use of the well-known triangle inequality and the inequalities (m + 1)! 2m , (κ )m κ m (m ∈ N) . We thus find

∞ ∞ m (−c/4) |c|m A[g p,b,c ](z) = z + ∑ zm+1 1 + ∑ m m=1 (m + 1)! (κ )m m=1 (m + 1)!4 (κ )m

|c| m−1 |c| ∞ = 1+ ∑ 8κ m=1 8κ

8κ |c| , κ> = . 8κ − |c| 8 2. Main results

T HEOREM 2.1. If the parameters b, p ∈ R, c ∈ C and, κ = p + (b + 1)/2 = 0, −1, −2, ... are so constrained that κ > 3|c| 8 , then g p,b,c (z) 8κ − 3 |c| (z ∈ U ) , ℜ 8κ − |c| g p,b,c n (z) and

ℜ

g p,b,c n (z) 8κ − |c| (z ∈ U ) . g p,b,c (z) 8κ + |c|

Proof. We observe from part (i) of Lemma 1.1 that 1+

∞

8κ + |c|

∑ |bm | 8κ − |c| ,

m=1

(2.1)

(2.2)

867


which is equivalent to 8κ − |c| ∞ ∑ |bm | 1, 2 |c| m=1 where bm =

(−c/4)m m!(κ )m .

Now, we may write g p,b,c (z) 8κ − |c| 8κ − 3 |c| − 2 |c| 8κ − |c| g p,b,c n (z) = :=

−|c| ∞ 1 + ∑nm=1 bm zm + 8κ2|c| ∑m=n+1 bm zm

1 + ∑nm=1 bm zm 1 + A(z) . 1 + B(z)

Set (1 + A(z))/ (1 + B(z)) = (1 + w(z)) / (1 − w(z)), so that w(z) = (A(z) − B(z)) / (2 + A(z) + B(z)). Then w(z) =

8κ −|c| 2|c|

m ∑∞ m=n+1 bm z

−|c| ∞ 2 + 2 ∑nm=1 bm zm + 8κ2|c| ∑m=n+1 bm zm

and |w(z)|

8κ −|c| 2|c|

∑∞ m=n+1 |bm |

−|c| ∞ 2 − 2 ∑nm=1 |bm | − 8κ2|c| ∑m=n+1 |bm |

.

Now |w(z)| 1 if and only if n 8κ − |c| ∞ |b | 2 − 2 m ∑ ∑ |bm | , |c| m=n+1 m=1

which is equivalent to n

∑ |bm | +

m=1

8κ − |c| ∞ ∑ |bm | 1. 2 |c| m=n+1

It suffices to show that the left hand side of (2.3) is bounded above by which is equivalent to 8κ − 3 |c| n ∑ |bm | 0. 2 |c| m=1

(2.3) 8κ −|c| 2|c|

∑∞ m=1 |bm | ,

868


To prove the result (2.2), we write 8κ + |c| 2 |c|

g p,b,c n (z) 8κ − |c| − g p,b,c (z) 8κ + |c|

−|c| ∞ 1 + ∑nm=1 bm zm − 8κ2|c| ∑m=n+1 bm zm

=

m 1 + ∑∞ m=1 bm z

:=

1 + w(z) 1 − w(z)

where |w(z)|

8κ +|c| 2|c|

∑∞ m=n+1 |bm |

−3|c| ∞ 2 − 2 ∑nm=1 |bm | − 8κ2|c| ∑m=n+1 |bm |

1.

The last inequality is equivalent to n

∑ |bm | +

m=1

8κ − |c| ∞ ∑ |bm | 1 2 |c| m=n+1

Since the left hand side of (2.4) is bounded above by completed.

8κ −|c| 2|c|

(2.4) ∑∞ m=1 |bm | , the proof is

T HEOREM 2.2. If the parameters b, p ∈ R, c ∈ C and, κ = p + (b + 1)/2 = 0, −1, −2, ... are so constrained that

κ>

3 |c| − 4 + 9 |c|2 + 8 |c| + 16 8

,

then

4κ (κ + 1) − (3κ + 2)|c| (z ∈ U ) , κ [4(κ + 1) − |c|]

(2.5)

g p,b,c n (z) κ [4(κ + 1) − |c|] (z ∈ U ) . ℜ g p,b,c (z) 4κ (κ + 1) + (κ + 2) |c|

(2.6)

ℜ and

gp,b,c (z) g p,b,c n (z)

Proof. From part (ii) of Lemma 1.1 we observe that 1+

∞

∑ (m + 1) |bm |

m=1

4κ (κ + 1) + (κ + 2)|c| , κ [4(κ + 1) − |c|]

which is equivalent to

κ [4(κ + 1) − |c|] ∞ ∑ (m + 1) |bm | 1, 2 (κ + 1)|c| m=1

869

G ENERALIZED B ESSEL FUNCTIONS (−c/4)m

where bm = m!(κ ) . m Now, we write

gp,b,c (z) κ [4(κ + 1) − |c|] 4κ (κ + 1) − (3κ + 2)|c| − 2 (κ + 1)|c| κ [4(κ + 1) − |c|] g p,b,c n (z) κ +1)−|c|] ∞ m 1 + ∑nm=1 (m + 1)bm zm + κ [4( 2(κ +1)|c| ∑m=n+1 (m + 1)bm z

=

1 + ∑nm=1 (m + 1)bmzm

:=

1 + w(z) , 1 − w(z)

where |w(z)|

κ [4(κ +1)−|c|] 2(κ +1)|c|

∑∞ m=n+1 (m + 1) |bm |

κ +1)−|c|] ∞ 2 − 2 ∑nm=1 (m + 1) |bm | − κ [4( 2(κ +1)|c| ∑m=n+1 (m + 1) |bm |

1.


∑ (m + 1) |bm | +

m=1

κ [4(κ + 1) − |c|] ∞ ∑ (m + 1) |bm | 1. 2 (κ + 1)|c| m=n+1

It suffices to show that the left hand side of (2.7) is bounded above by ∑∞ m=1 (m + 1) |bm | , which is equivalent to

(2.7) κ [4(κ +1)−|c|] 2(κ +1)|c|

4κ (κ + 1) − (3κ + 2)|c| n ∑ (m + 1) |bm | 0. 2 (κ + 1)|c| m=1 To prove the result (2.6), we write 4κ (κ + 1) + (κ + 2) |c| 2 (κ + 1)|c| :=

g p,b,c n (z) κ [4(κ + 1) − |c|] − gp,b,c (z) 4κ (κ + 1) + (κ + 2) |c|

1 + w(z) , 1 − w(z)

where |w(z)|

4κ (κ +1)+(κ +2)|c| ∞ ∑m=n+1 (m + 1) |bm | 2(κ +1)|c| 4κ (κ +1)−(3κ +2)|c| ∞ n 2 − 2 ∑m=1 (m + 1) |bm | − ∑m=n+1 (m + 1) |bm | 2(κ +1)|c|

1.


∑ (m + 1) |bm | +

m=1

κ [4(κ + 1) − |c|] ∞ ∑ (m + 1) |bm | 1. 2 (κ + 1)|c| m=n+1

Since the left hand side of (2.8) is bounded above by the proof is completed.

κ [4(κ +1)−|c|] 2(κ +1)|c|

(2.8)

∑∞ m=1 (m + 1) |bm | ,

870


T HEOREM 2.3. If the parameters b, p ∈ R, c ∈ C and, κ = p + (b + 1)/2 = 0, −1, −2, ... are so constrained that κ > |c| 4 , then A[g p,b,c ](z) 8κ − 2 |c| (z ∈ U ) , ℜ 8κ − |c| A[g p,b,c ] n (z) and

(2.9)

A[g p,b,c ] n (z) 8κ − |c| ℜ (z ∈ U ) , A[g p,b,c ](z) 8κ

(2.10)

where A[g p,b,c ] is the Alexander transform of g p,b,c . Proof. We prove only (2.9), which is similar in spirit to the proof of Theorem 2.1. The proof of (2.10) follows the pattern of that in (2.2). We consider from part (iii) of Lemma 1.1 that 1+ which is equivalent to

∞

8κ |bm | , m + 1 8 κ − |c| m=1

∑

8κ − |c| ∞ |bm | ∑ m + 1 1, |c| m=1

m

where bm = (−c/4) . m!(κ )m We may write

A[g p,b,c ](z) 8κ − |c| 8κ − 2 |c| − |c| 8κ − |c| A[g p,b,c ] n (z) = :=

−|c| ∞ bm m bm m 1 + ∑nm=1 m+1 z + 8κ2|c| z ∑m=n+1 m+1 bm m 1 + ∑nm=1 m+1 z

1 + w(z) , 1 − w(z)

where |w(z)|

8κ −|c| ∞ bm m 2|c| ∑m=n+1 m+1 z |bm | −|c| ∞ |bm | 2 − 2 ∑nm=1 m+1 − 8κ|c| ∑m=n+1 m+1

1.

The last inequality is equivalent to 8κ − |c| ∞ |bm | |bm | + ∑ m + 1 1. |c| m=n+1 m=1 m + 1 n

∑

It suffices to show that the left hand side of (2.11) is bounded above by which is equivalent to 8κ − 2 |c| n |bm | 0. ∑ |c| m=1 m + 1

(2.11) 8κ −|c| |c|

|b |

m ∑∞ m=1 m+1 ,

871


2.1. Bessel functions Choosing b = c = 1, in (1.3) or (1.4), we obtain the Bessel function J p (z) of the first kind of order p defined by (1.5). Let J p : U −→ C be defined by p √ J p (z) = 2 p Γ(p + 1)z1− 2 J p ( z). We observe that

√ √ √ √ √ 3 sin z J−1/2 (z) = z cos z, J1/2 (z) = z sin z, J3/2 (z) = √ − 3 cos z. z

In particular, the results of Theorems 2.1-2.3 become: C OROLLARY 2.4. The following assertions hold true: (i) If p > − 58 , then J p (z) 8p + 5 ℜ (z ∈ U ) , (J p )n (z) 8p + 7 and

(ii) If p >

√ −9+ 33 8

(J p )n (z) ℜ J p (z)

8p + 7 (z ∈ U ) . 8p + 9

≈ −0.406 93 then J p (z) 4p2 + 9p + 3 ℜ (z ∈ U ) , 2 4p + 11p + 7 J p n (z)

and ℜ

J p n (z) J p (z)

4p2 + 11p + 7 (z ∈ U ) . 4p2 + 13p + 11

(iii) If p > − 34 , then A [J p ] (z) 8p + 6 (z ∈ U ) , ℜ (A [J p ])n (z) 8p + 7 and

ℜ

(A [J p ])n (z) A [J p ] (z)

8p + 7 (z ∈ U ) . 8p + 8

(2.12)

(2.13)

(2.14)

(2.15)

(2.16)

(2.17)

√ R EMARK 2.5. For p = −1/2 we get J−1/2 (z) = z cos z, and for n = 0, we have J−1/2 0 (z) = z, so,

and

√ 1 (z ∈ U ) , ℜ cos z 3

(2.18)

√ 3 ℜ 1/ cos z (z ∈ U ) . 5

(2.19)

872


R EMARK 2.6. If we take p = 1/2 we have J1/2 (z) = and for n = 0, we get J1/2 0 (z) = z, so,

√ √ √ sin z+ z cos z √ , 2 z

√ √ (z) = z sin z and J1/2

√ sin z 9 ≈ 0, 82 (z ∈ U ) , √ ℜ z 11

(2.20)

√ z 11 ℜ √ ≈ 0, 85 (z ∈ U ) , sin z 13 √ √ sin z 34 ℜ ≈ 1, 26 (z ∈ U ) , √ + cos z z 27 and

(2.21) (2.22)

√ z 27 ℜ √ √ √ ≈ 0, 36 (z ∈ U ) . sin z + z cos z 74

R EMARK 2.7. For p = 3/2 , we have J3/2 (z) = 3 √ √

(z−1) sin z 3 cos z √ and J3/2 0 (z) = z. Hence, 2 z + z z ℜ

√ sin z √ z

(2.23)

√

(z) = − cos z , J3/2

√ √ sin z cos z 17 ≈ 0, 30 (z ∈ U ) , √ − z z z 57

(2.24)

√ z z 19 √ √ √ ≈ 2, 71 (z ∈ U ) , sin z − z cos z 7 √ √ cos z (z − 1) sin z 102 ℜ + √ ≈ 0, 52 (z ∈ U ) , z z z 195 ℜ

and

(2.25) (2.26)

√ z z 195 ≈ 1, 23 (z ∈ U ) . ℜ √ √ √ z cos z + (z − 1) sin z 158

(2.27)

2.2. Modified Bessel functions Taking b = 1 and c = −1, in (1.3) or (1.4), we obtain the modified Bessel function I p (z) of the first kind of order p defined by (1.6). Let the function I p : U −→ C be defined by p √ I p (z) = 2 p Γ(p + 1)z1− 2 I p ( z). We observe that √ √ √ √ √ 3 sinh z . I−1/2 (z) = z cosh z, I1/2 (z) = z sinh z, I3/2 (z) = 3 cosh z − √ z The properties of the function I p are the same like for the function J p , because in this case we have |c| = 1.More precisely, we have the following results.

873


C OROLLARY 2.8. The following assertions are true: (i) If p > − 58 , then

I p (z) ℜ (I p )n (z) and

8p + 5 (z ∈ U ) , 8p + 7

(2.28)

8p + 7 (z ∈ U ) . 8p + 9

(2.29)

4p2 + 9p + 3 (z ∈ U ) , 4p2 + 11p + 7

(2.30)

4p2 + 11p + 7 (z ∈ U ) . 4p2 + 13p + 11

(2.31)

(I p )n (z) ℜ I p (z) (ii) If p >

√ −9+ 33 8

and ℜ

≈ −0.406 93 then

ℜ

I (z) p I p n (z)

I p n (z) I p (z)

(iii) If p > − 34 , then

A [I p ] (z) ℜ (A [I p ])n (z) and

(A [I p ])n (z) ℜ A [I p ] (z)

8p + 6 (z ∈ U ) , 8p + 7

(2.32)

8p + 7 (z ∈ U ) . 8p + 8

(2.33)

√ R EMARK 2.9. For p = −1/2 we get I−1/2 (z) = z cosh z, and for n = 0, we have I−1/2 0 (z) = z, so,

and

√ 1 ℜ cosh z (z ∈ U ) , 3

(2.34)

√ 3 ℜ 1/ cosh z (z ∈ U ) . 5

(2.35)

R EMARK 2.10. If we take p = 1/2 we have I1/2 (z) = √ √ √ sinh z+ z cosh z √ , and for n = 0, we get I1/2 0 (z) = z, so, 2 z

√ sinh z √ z √ z ℜ √ sinh z ℜ

√ √ z sinh z and I1/2 (z) =

9 ≈ 0, 82 (z ∈ U ) , 11

(2.36)

11 ≈ 0, 85 (z ∈ U ) , 13

(2.37)

874


√ √ sinh z 34 + cosh z ≈ 1, 26 (z ∈ U ) , ℜ √ z 27 and

ℜ

√ z 27 √ √ √ ≈ 0, 36 (z ∈ U ) . sinh z + z cosh z 74

(2.38)

(2.39)

√

√ sinh z (z) = R EMARK 2.11. For p = 3/2 , we have I3/2 (z) = 3 cosh z − √z , I3/2 √ √

sinh z cosh z 3 (1+z)√ − z and I3/2 0 (z) = z. Hence, 2 z z

√ √ cosh z sinh z 17 − √ ≈ 0, 30 (z ∈ U ) , ℜ z z z 57 √ z z 19 ℜ √ √ √ ≈ 2, 71 (z ∈ U ) , z cosh z − sinh z 7 √ √ (1 + z) sinh z cosh z 102 √ − ≈ 0, 52 (z ∈ U ) , ℜ z z z 195 and

(2.40) (2.41) (2.42)

√ z z 195 ≈ 1, 23 (z ∈ U ) . ℜ √ √ √ (1 + z) sinh z − z cosh z 158

(2.43)

2.3. Spherical Bessel functions If we take b = 2 and c = 1, in (1.3) or (1.4), we obtain the spherical Bessel function S p (z) of the first kind of order p defined by (1.7). C OROLLARY 2.12. Let S p : U −→ C be defined by p √ S p (z) = 2 p Γ(p + 1)z1− 2 S p ( z).

Then the following assertions are true: (i) If p > − 98 , then ℜ and

ℜ (ii) If p >

√ −13+ 33 8

ℜ

S p (z) (S p )n (z) (S p )n (z) S p (z)

8p + 9 (z ∈ U ) , 8p + 11

(2.44)

8p + 11 (z ∈ U ) . 8p + 13

(2.45)

≈ −0.906 93 then S (z) p Sp n (z)

8p2 + 26p + 17 (z ∈ U ) , 8p2 + 30p + 27

(2.46)

875


and ℜ

Sp n (z) Sp (z)

8p2 + 30p + 27 (z ∈ U ) . 8p2 + 34p + 37

(2.47)

(iii) If p > − 54 , then ℜ and

A [S p ] (z) (A [S p ])n (z)

(A [S p ])n (z) ℜ A [S p ] (z)

8p + 10 (z ∈ U ) , 8p + 11

(2.48)

8p + 11 (z ∈ U ) . 8p + 12

(2.49)

3. Illustrative examples and image domains In this section, we present several illustrative examples along with the geometrical descriptions of the image domains of the unit disk by the ratio of Bessel (modified Bessel) function to its sequence of partial sums or the ratio of its sequence of partial sums to the function which we considered in our remarks in section 2. √ E XAMPLE 3.1. The image domains of f1 (z) = cos z and f2 (z) = shown in Figure 1.

Figure 1.

1√ cos z

are

876


E XAMPLE 3.2. We present the image domains of f3 (z) = √ √ sin z and f5 (z) = √z + cos z in Figure 2.

√ sin z √ , z

f4 (z) =

√ z √ sin z

Figure 2.

E XAMPLE 3.3. We have the image domains of f6 (z) =

√ cos z z

+

√ (z−1) sin z √ z z

√ √ sin z cos z √ − z z z

and f7 (z) =

in Figure 3.

Figure 3.

E XAMPLE 3.4. The image domains of f8 (z) = and f10 (z) =

√ z z √ √ √ (1+z) sinh z− z cosh z

1√ , cosh z

are shown in Figure 4.

Figure 4.

f9 (z) =

√ sinh z √ z

√ + cosh z


877

Acknowledgement. The present investigation was supported by Ataturk University Rectorship under ”The Scientific and Research Project of Atatürk University”, Project No: 2012/173. REFERENCES [1] J. W. A LEXANDER, Functions which map the interior of the unit circle upon simple regions, Ann. of Math. 17 (1915), 12–22. [2] A. BARICZ, Geometric properties of generalized Bessel functions, Publ. Math. Debrecen 73, 1–2 (2008), 155–178. [3] A. BARICZ, Functional inequalities involving special functions, J. Math. Anal. Appl. 319 (2006), 450–459. [4] A. BARICZ, Functional inequalities involving special functions. II, J. Math. Anal. Appl. 327 (2007), 1202–1213. [5] A. BARICZ, Some inequalities involving generalized Bessel functions, Math. Inequal. Appl. 10 (2007), 827–842. [6] A. BARICZ AND S. P ONNUSAMY, Starlikeness and convexity of generalized Bessel functions, Integral Transforms Spec. Funct. 21 (2010), 641–653. [7] L. B RICKMAN , D.J. H ALLENBECK , T.H. M AC G REGOR AND D. W ILKEN, Convex hulls and extreme points of families of starlike and convex mappings, Trans. Amer. Math. Soc. 185 (1973), 413– 428. [8] E. D ENIZ , H. O RHAN AND H.M. S RIVASTAVA, Some sufficient conditions for univalence of certain families of integral operators involving generalized Bessel functions, Taiwanese J. Math. 15, 2 (2011), 883–917. [9] E. D ENIZ, Convexity of integral operators involving generalized Bessel functions, Integral Transforms Spec. Funct. 1 (2012), 1–16. [10] L.J. L IN AND S. O WA, On partial sums of the Libera integral operator, J. Math. Anal. Appl. 213, 2 (1997), 444–454. [11] H. O RHAN AND E. G UNES , Neighborhoods and partial sums of analytic functions based on Gaussian hypergeometric functions, Indian J. Math. 51, 3 (2009), 489–510. [12] S. O WA , H.M. S RIVASTAVA AND N. S AITO, Partial sums of certain classes of analytic functions, Int. J. Comput. Math. 81, 10 (2004), 1239–1256. [13] T. S HEIL -S MALL, A note on partial sums of convex schlicht functions, Bull. London Math. Soc. 2 (1970), 165–168. [14] H. S ILVERMAN, Partial sums of starlike and convex functions, J. Math. Anal. Appl. 209 (1997), 221– 227. [15] E.M. S ILVIA, On partial sums of convex functions of order α , Houston J. Math. 11 (1985), 397–404. [16] G. N. WATSON, A Treatise on the Theory of Bessel Functions, Second edition, Cambridge University Press, Cambridge, London and New York, 1944.

(Received July 19, 2013)

Halit Orhan Department of Mathematics, Faculty of Science Atatürk University Erzurum, 25240, Turkey e-mail: [email protected] Nihat Yagmur Department of Mathematics, Faculty of Science and Art Erzincan University Erzincan, 24000, Turkey e-mail: [email protected]

Journal of Mathematical Inequalities

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