HARMONIC UNIVALENT FUNCTIONS INVOLVING FOX - WRIGHT

  Bulletin of Mathematics

  ISSN Printed: 2087-5126; Online: 2355-8202 Vol. 08, No. 02 (2016), pp.133-141 http://jurnal.bull-math.org

  

HARMONIC UNIVALENT FUNCTIONS INVOLVING

FOX - WRIGHT

AIBAH

  IREGAR ORHEZAN MAR AND N ZMAR N AUD S S , N U T . A T .D

  Abstract. In this paper we introduce the new subclass of analytic and harmonic univalent func- tions involving Fox-Wright functions.The coefficient bounds and Growth Theorem as well as Dis- tortion Theorem results for these functions are obtained.

1. INTRODUCTION

  A continuous function f (u, v) = u + iv is a complex-valued harmonic function in a simply connected complex domain C if both u and v are real harmonic in

C. Let H denote the family of function f = h + g which are harmonic and sense

• preserving in the open unit disk U = z : |z| < 1 where h and g are given by

  ∞ ∞ k k , h (z) = z + a k z g (z) = b k z (1)

  ∑ ∑

  k =2 k =1 The function h is called the analytic part and g is called the co-analytic part of the harmonic function f = h + g. The class H reduces to the class of normalized univalent analytic functions, if the co-analytic part of f is zero. A necessary and sufficient condition for f in H to be locally univalent and sense-preserving in U

  ′ ′( is that h (z) > g z ) in U . Let T H be the class of functions in H that may be expressed as f

  = h + g where Received 06-12-2016, Accepted 26-12-2016.

  2010 Mathematics Subject Classification : 30C45 Key words and Phrases : Harmonic, Univalent, starlike, Fox-Wright functions.

  Siregar, S. et al. – Harmonic Univalent Function Involving Fox - Wright

  ∞ ∞ k k h a z , g b z , a (2)

  (z) = z − k (z) = k k ≥ 0, b k ≥ 0

  ∑ ∑

  k =2 k =1 Let S

  H (α) and K H (α) be the subclasses of H consisting of univalent harmonic functions starlike of order and convex of order α, respectively, where 0 ≤ α < 1 and

  |z| = r < 1, if ∂ i

  θ (arg f (re )) ≥ α, |z| = r < 1, (3)

  ∂ θ and " #

  ∂ ∂ i θ arg f (re ) ≥ α, |z| = r < 1, (4)

  ∂ θ ∂ θ Also let T S (α) and T K (α) be the respective subclasses of S (α) and K (α)

  H H H H consisting of functions of the form (2).

  Harmonic functions are indeed famous for their use in the study of minimal sur- faces and play an important role in a variety of problems in applied mathematics. Harmonic functions have been studied by different geometers such as Kneser [8]. In 1984, Clunie and Sheil-Small [2] began a study of complex-valued, harmonic mappings defined on a domain U ⊂ C. The coefficients bounds for the classes S H (α), K H (α), T S H (α), T K H (α) a are stud- ied in details by Silverman [11], Jahangiri [4], Jahangiri & Silverman [5], Silverman & Silvia [12],Jakubowski et al [6], Janteng et. al [7], Yalcin, [13], [14], [15] and Siregar et.al [10].

2. PRELIMINARY RESULTS

  In this paper, we considered the function as follows ∞

  α m m k Ω Θ

  I (α , β ) f (z) = z + a z , (5)

  1 1 k m,k ∑ k k k =2

      q

  Γ α j + A j (k − 1)

   

  ∏

     

   j =1 

  1 m

    Ω   ,

  = (6) k  s 

   

  (k − 1)!  

    Γ

  β j + B j (k − 1)  

  ∏

  j =1 " # k

  (k − 1) (k + λ − 2)! m

  Θ . (7) = k

  λ !(k − 2)! Siregar, S. et al. – Harmonic Univalent Function Involving Fox - Wright

  In this section, the classes S H (Ω, Θ, γ) and T H (Ω, Θ, γ) , of functions which are harmonic univalent in U will be introduced. Some properties for functions f belonging to these classes which include the coefficient estimates, growth results and distortion theorem will be given.

  Let f = h + g denote in the form (2). f be in class function of S (Ω, Θ, γ) if H satisfy the condition

    ∂ λ

    (I f (z)) m,k

  ∂ θ i θ Re > 1 −|γ| , (z = re , z ∈ U) (8)

  ∂  z 

  ∂ θ α α where, I f f

  (x) in (1.5), 0 < |γ| ≤ 1 and I (z) = I(h) + I(g(z)) m,k m,k

3. COEFFICIENT BOUNDS

  Theorem 1. Let f = h + g with h and g given by (2), ∞ m m

  Ω Θ k (|a | +|b |) ≤ |γ| −|b | , (9) k k

  1

  ∑ k k

  k =2

      q

  Γ  α + A (k − 1)  j j

  ∏

      h k i

   j  =1 (k−1) (k+λ −2)! m

  1 m   where Ω   , Θ . Then

  = = s k   k

  (k−1)! λ !(k−2)!  

     

  Γ β j + B j (k − 1)

   ∏  j

  =1 f is harmonic univalent sense preserving in U and f ∈ S H (Ω, Θ, γ).

  ∞ ∞ k k Proof. Let f = h + g where ,h(z) = z − ∑ |a k | z and g (z) = ∑ |b k | z , |b 1 | < 1. k k

  =2 =1 For |z | ≤ |z | < 1, it sufficient to show that, f (z ) − f (z ) >

  0. So,

  1

  2

  1

  2 f h ,

  (z 1 ) − f (z 2 ) = (z 1 ) + g(z 1 ) − h(z 2 ) + g(z 2 ) = h (z ) − h(z ) − g(z ) − g(z )

  1

  2

  1

  2 By using triangle inequality for the above function is, f (z 1 ) − f (z 2 ) ≥ h (z 1 ) − g(h 2 ) − g (z 1 ) − g(z 2 ) (10)

  Substitute the function h and g from (2) respectively, into equation (10) become

  Siregar, S. et al. – Harmonic Univalent Function Involving Fox - Wright f (z 1 ) − f (z 2 ) ≥ z

  k

  ||z| k −1

  k =2 k |a k

  ∑

  ∞

  ′ (z) ≥ 1 −

  Consequently, f is univalent in U . To prove that f is sense preserving in U . This is because h

  , in equation (6) and (7) respectively. f (z 1 ) − f (z 2 ) ≥ |z 1 − z 2 | (1 −|b 1 | −|z 2 ||γ| −|b 1 |) > 0

  (k+λ −2)! λ !(k−2)! i

  = h (k−1)

  ∑

  , Θ m k

        

  1 (k−1)!

  β j + B j (k − 1)      

  j =1 Γ

  ∏

  Γ α j + A j (k − 1) s

  j =1

  ∏

  > 1 − ∞

  k =2 k |a k

  =       

  |b k | , >

  ′ (z) ≥ g

  (12) ∴ h

  (z) ,

  ≥ g ′

  ||z| k −1

  k =1 k |b k

  ∑

  ∞

  ] γ

  | , take U = {z : |z| < 1, z ∈ C} ≥ 1 −

  Θ m k

  k =1 k [Ω m k

  ∑

  |a k | ≥ 1 − ∞

  ] γ

  Θ m k

  k =2 k [Ω m k

  ∑

  ∞

        q

  , where Ω m k

  1 − ∞ ∑ k =2 a k z k

  1 − z k

  ∑

  ∞

  2 ) +

  1 − z

  2 ) ≥ (z

  1 ) − f (z

  using the properties of sigma, from equation (11), we obtain f (z

  2 (11)

  2 ∞ ∑ k =1 b k z k

  (z k

  1 − z k

  ∞ ∑ k =2 a k z k

  2 ^! = (z 1 − z 2 ) +

  2 − ∞ ∑ k =1 b k z k

  1 ^! − z

  1 − ∞ ∑ k =1 b k z k

  2 ^! z

  2 −

∞

∑

k

=2

a k z k

  1 ^! − z

  k =2 a k

  1 − z k

  [|a k | +|b k |] !

  ! . From the equation in (5), will be finding f

  Θ k m i

  Ω k m

  k =2 k h

  ∑

  ∞

  1 −|b 1 | −

  2

  (z 1 ) − f (z 2 ) f (z 1 ) − f (z 2 ) ≥ absz 1 − z

  k =2 k [|a k | +|b k |]|z n | k −1

  2 ) −|b

  ∑

  ≥ |z 1 − z 2 | 1 −|b 1 | − ∞

  2 ),

  1 − z k

  (z k

  k =1 a k

  ∑

  ∞

  1 | +

  ′ (z) . Therefore f is sense preserving in U .

• ∞
• ∞

  Θ m k

  | !

  ]|b k

  Θ m k

  k =1 k [Ω m k

  

∑

  | − ∞

  ]|a k

  Θ m k

  k =2 k [Ω m k

  ∑

  |γ| − ∞

    

  −1 ≥ 2

  ]b k z k

  k =1 k [Ω m k

  ≥ 0 The harmonic mappings f (z) = z +

  ∑

  ∞

  −1 − 2

  ]a k z k

  Θ m k

  k =2 k [Ω m k

  ∑

  ∞

  −1 = 2|γ| − 2

  ]b k z k

  Θ m k

  k =1 k [Ω m k

  ∑

  −1

    

  ∞

  Θ m k

  k =1 k

  |y k

  k =1

  ∑

  | + ∞

  |x k

  k =2

  

∑

  |) = ∞

  | +|b k

  (|a k

  1

  ]

  Θ m k

  [Ω m k

  ∑

  ∑

  ] z k

  k =2 |γ| x k k [Ω m k

  Θ m k

  ] z k

  ∑

  k =1 |γ| y k k [Ω m k

  Θ m k

  (13) where ∞

  ∞

  ∑

  k =2 |x n | +

  ∞

  ∑

  k =1 |y n | = 1, it shows that the coeddicient bound given by (9) is sharp. The functions of form (13) are in S

  H (Ω, Θ, γ) because

  ]a k z k

  k =2 k [Ω m k

  | = 1 The restriction placed in Theorem 1 on the moduli of the coefficients of f = h + g enables us to conclude for arbitrary rotation of the coefficients of f that the resulting

  So, the I λ m,k

  = γ + 1 + ∞

  λ m,k g (z))

  − 2 − γ − (I λ m,k h (z)) + (I

  λ m,k g (z))

  λ m,k h (z)) − (I

  , ⇔ γ + (I

  , it suffieces to show that |γ| + w ≥ 2 −|γ| − w

  2 −|γ| − w

  β 1 )g(z), become Now f ∈ S H (Ω, Θ, γ). Usung the fact that Re{w} > 1 −|γ| if and only if |γ| + w ≥

  1 ,

  (α

  β 1 )h(z) and I λ m,k

  1 ,

  (α

  1 )g(z)

  k =2 k [Ω m k

  1 , β

  λ m,k (α

  1 )h(z) + I

  1 , β

  λ m,k (α

  1 ) f (z) = I

  1 , β

  (α

  I λ m,k

  β 1 ) f (z) in equation (5),

  1 ,

  (α

  Next I λ m,k

  Siregar, S. et al. – Harmonic Univalent Function Involving Fox - Wright

  ∑

  Θ m k

  ∑

  ]b k z k

  ∞

  −1 −

  ]b k z k

  Θ m k

  k =1 k [Ω m k

  ∑

  ∞

  −1 −

  ]a k z k

  Θ m k

  k =2 k [Ω m k

  ∑

  ∞

  −1 ≥ 2|γ|−

  Θ m k

  ]a k z k

  k =1 k [Ω m k

  −1 −

  ∞

  ∑

  k =1 k [Ω m k

  Θ m k

  ]b k z k

  −1 − 2 − γ − 1 −

  ∞

  ∑

  k =2 k [Ω m k

  Θ m k

  ]a k z k

  −1

  ∑

• ∞

• |γ|

Siregar, S. et al. – Harmonic Univalent Function Involving Fox - Wright

  Next the condition (10) is also necessary for functions f to be in T H (Ω, Θ, γ). Theorem 2. Let f

  = h + g with h and g given by (2). Then f ∈ T H (Ω, Θ, γ) if and only if the inequality (9) holds for the coefficients of f = h + g. Proof. First suppose that f

  ∈ T H (Ω, Θ, γ), then by (8) have n o λ ′

  λ Re (I h (z)) − I g (z) m,k m,k

  ( ) ∞ ∞ m m k m m k

  −1 −1 Θ Θ >

  = Re 1 − k [Ω ]a z − k [Ω ]b z 1 −|γ| k k

  ∑ k k ∑ k k

  k k =2 =1

  ∞ m m −

  Θ If choose z to be real and let z → 1 , then we can have 1 − k [Ω ]|a | − k

  ∑ k k

  k =2 ∞ m m

  Θ k [Ω ]|b k | > 1 −|γ|, which is precisely the assertion (9). Conversely, suppose k k

  ∑

  k =1 that the inequality (9) holds true. Then can be find from the equation (8) that n o

  λ ′ λ

  Re (I h (z)) − I g (z) m,k m,k

  ( ) ∞ ∞ m m k m m k

  −1 −1 Θ Θ

  = Re 1 − k [Ω ]a z − k [Ω ]b z k k

  

∑ k k ∑ k k

  k k =2 =1

  ∞ m m k −1 Θ

  ≥ 2 − k [Ω ](|a k | +|b k |)|z|

  ∑ k k

  k =1 ∞ m m

  > 2 k Θ − [Ω ](|a k | +|b k |) ≥ 1 −|γ|

  ∑ k k

  k =1 provided that the inequality (9) is satisfied.

4. GROWTH BOUNDS AND DISTORTION THEOREM

  In this subsection, growth bounds for functions in T H (Ω, Θ, γ) will be obtained and extreme points for this class will be given. Theorem 3. If f ∈ T (Ω, Θ, γ) for 0 < |γ| ≤ 1, N , λ ≤ 0 and |z| = r > 1, then

  H |γ| −|b 1 |

  2 f (z) ≤ (1 +|b |) r + r

  1 m m 2 Θ

  [Ω ] k k and

  |γ| −|b 1 |

  2 f (z) ≥ (1 −|b 1 |) r − r m m

  2 Θ [Ω ] k k Siregar, S. et al. – Harmonic Univalent Function Involving Fox - Wright

  Proof. Let f ∈ T H (Ω, Θ, γ). Taking the absolute value of f (z). The right hand side f (z) ≥ (1 +|b

  ∑

  [|a k | +|b k |]r

  1 |

  ] |γ| −|b

  Θ m k

  2 [Ω m k

  k =2

  2 ∞

  (z) ≤ (1 +|b 1 |) r + |γ| −|b

  ] r

  Θ m k

  |γ| −|b 1 | 2 [Ω m k

  2 ≤ (1 +|b 1 |) r +

  k =2 [|a k | +|b k |]r

  ∑

  2 And also, by equation (9), we find f

  1 | 2 [Ω m k

  |]r k

  2 Θ m

  2 Θ m

  Ω m

  1 |

  |γ| −|b

  ′ (z) ≥ (1 −|b 1 |) −

  2 r, and f

  Ω m

  Θ m k

  1 |

  (z) ≤ (1 +|b 1 |) + |γ| −|b

  (Ω, Θ, γ) for 0 < |γ| ≤ 1, m ∈ N , λ ≥ 0 and |z| = r > 1 f ′

  Theorem 4. If f ∈ T H

  Distortion for function in T H (Ω, Θ, γ) will be obtained by Theorem 4.

  2 (15) The proof is complete.

  ] r

  ≤ (1 +|b 1 |) r + ∞

  | +|b k

  1 |) r −

  ∑

  ] ∞

  Θ m k

  2 [Ω m k

  |γ| −|b 1 |

  2 ≥ (1 +|b 1 |) r −

  k =2 [|a k | +|b k |]r

  ≥ (1 +|b 1 |) r − ∞

  k =2

  |]r k

  | +|b k

  [|a k

  k =2

  ∑

  ∞

  ∑

  2 [Ω m k

  [|a k

  ] r

  k =2

  ∑

  ∞

  1 |) r +

  The left hand side, f (z) ≤ (1 +|b

  2 (14)

  Θ m k

  Θ m k

  1 | 2 [Ω m k

  |γ| −|b

  2 By equation (9), we obtain f (z) ≥ (1 +|b 1 |) r −

  |]r

  | +|b k

  [|a k

  ] |γ| −|b 1 |

  2 r. Siregar, S. et al. – Harmonic Univalent Function Involving Fox - Wright

  To obtained distortion theorem, it can be differentiate the equation in (14) and (15), then

  |γ| −|b |

  1 ′ f r,

  (z) ≤ (1 +|b 1 |) + m m

  Ω Θ

  2

  2 and

  |γ| −|b 1 | ′ f (z) ≥ (1 −|b 1 |) − r. m m

  Ω Θ

  2

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13. Yalcin, S. & O”ztu”rk, M. 2006. On a subclass of certain convex harmonic

  Siregar, S. et al. – Harmonic Univalent Function Involving Fox - Wright

  

14. Yalcin, S., O”ztu”rk, M. & Yamankaradeniz, M. 2000. On some subclasses of

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  Saibah Siregar : Department of Science and Biotechnology, Faculty of Engineering and Life Sciences, University of Selangor, Bestari Jaya 45600, Selangor D.E. Malaysia

  E-mail: saibah@unisel.edu.my

  Norhezan Umar : Department of Science and Biotechnology, Faculty of Engineering and Life Sciences, University of Selangor, Bestari Jaya 45600, Selangor D.E. Malaysia

  E-mail: norhezanumar@unisel.edu.my

  Tn. Azmar Tn.Daud : Department of Science and Biotechnology, Faculty of Engineering

and Life Sciences, University of Selangor, Bestari Jaya 45600, Selangor D.E. Malaysia

  E-mail: azmardaud@unisel.edu.my

  Siregar, S. et al. – Harmonic Univalent Function Involving Fox - Wright .