Properties for an integral operator of p-valent functions Текст научной статьи по специальности «Математика»

YflK 517.44

Вестник СПбГУ. Сер. 1. Т. 1 (59). 2014. Вып. 3

PROPERTIES FOR AN INTEGRAL OPERATOR OF p-VALENT FUNCTIONS

Nicoleta Ularu1, Laura Stanciu2

1 "loan Slavici" University of Timigoara,

str. Aurel Podeanu, No. 144, Timigoara, Romania

2 University of Pitesti,

str. Targul din Vale, No. 1, Pitesti Romania

The purpose of this paper is to obtain new sufficient conditions for an operator on the classes of starlike and convex functions of order a and type a, the class of p-valently starlike functions, p-valently convex functions and uniformly p-valent starlike and convex functions. Refs 12.

Keywords: Analytic functions, close-to-convex functions, close-to-starlike functions, integral operator.

1. Introduction and definitions

Let Ap the class of all p-valent analytic functions

f(z) = zp + ap+1zp+1 + ...,p e N

on the open unit disk U = {z e C : \z\ < 1}. If we consider p =1 we obtain that A1 = A, the class of all analytical functions on U that satisfy the condition f (0) = f '(0) — 1 = 0. We consider the classes introduced and studied by R. M.Ali and V. Ravichandran

in [1].

The class of p-valent starlike functions is denoted by S*(y) and satisfy the condition

ifte (ZJM) > 7

pKe{f(z) J > 7

for y < 1 and z eU.

A functions f e Ap is in the class of p-valent convex functions if

1 (zf''(z) \

and we denote this class by Kp.

Starting from the classes of starlike and convex functions of complex order a and type a, R. Ali and V. Ravichandran in [1] defined the classes S*(a, a) and Kp(a, a) as follows:

S*p (a, a) = j/ G Ap, a < 1 : Re + -j- Q^y - > a

and

= 1 :Re^l + i Q + >a

In the case of p =1 the classes were studied by Breaz [4], Frasin [6], etc. Next we will consider the classes Mp(y) and Np(y).

A function f e Ap is in the class Mp(y) if

iRe (zJm < 7

P l/wJ 7

for y > 1.

The class Np(y) contains all the functions that satisfy the condition

1 (zf"(z) \

;Re (tw + V4 7

for f e Ap and y > 1-

If we consider p = 1, we obtain the classes M(y) and N(y) that were studied by many others, for example: Breaz [3], Ularu, Breaz and Frasin in [12] and Uralegaddi et al. in [11].

Also they have defined in a analogue mode the classes Mp(a, a) and Np(a, a). A function f e Ap is in the class Mp(a, a) if

for a > 1.

The class Np(a, a) contains all the functions f eAp that satisfy for a > 1.

A function f is uniformly p-valent starlike of order a with —1 < a < p in the open unit disk if

I /(*) J ~ f

for z eU. This class was introduced by Goodman in [7].

The class of uniformly p-valent close-to-convex functions of order a with 0 < a < p in U contains all the functions that satisfy

—p

Re ( fffl _ a ] >

g(z)

zf'(z)

wr~p

for z eU and the function g from the class of p-valent starlike functions of order a.

To prove that our functions are p-valently starlike and p-valently close-to-convex in the open unit disk we will use the following lemmas:

Lemma 1.1 (8). If f eAp satisfies

zf"(z) 1 MI + ^TtVI <i?+7 forzeU, (1.1)

f'(z) 1 4

then f is p-valently starlike in U.

Lemma 1.2 (5). If f eAp satisfies

zf"(z)

+ 1 — p

<p +1 for z eU, (1.2)

f '(z)

then f is p-valently starlike in U. Lemma 1.3 (9). If f eAp satisfies

zf''(z\ a + b , N

for z eU, where a > 0,b > 0 and a + 2b < 1, then f is p-valently close-to convex in U. Lemma 1.4. [2] If f eAp satisfies

Refl + W1<,, + 5 (L4)

for z eU, then f is uniformly p-valent close-to-convex in U. To prove our results we consider the integral operator

Gp,n(z)= /V1!!!^) dt> (1.5)

(Mt)\(gi(t)

/0 --V t ) \ptp-1

that was studied by Stanciu and Ularu in [10]. 2. Main results

Theorem 2.1. Let fi,gi g Ap, /Xj, Aj > 0 and on,$i < 1 for i = l,n. If fi g <5*(/3j) and

n n

gi e Kp(ai), then Gp,n e Kp(Y), where y =1 — J2 Mi(1 — Pi) — E Ai(1 — a).

i=i i=i

Proof. Starting from relation (1.5) and by logarithmic differention we obtain that It follows that

1 , zGvAz)\ _ 1 („ , fzfl(z) , ^

and

^J, , ;G';,«(;)\ni.. (¿KM . > (zgUz)

Using that fi e S*(Pi) and gi e Kp(ai), we obtain

M' + frr

p V G'p,n(z) / i=1

Re (1 + ^fl) > 1 - f>( 1 - A) " E Aj(l - a,) = 7.

V Gp,n(z) / i=i i = i

□

For n = p = 1 in Theorem 2.1 we obtain Corollary 2.2. Let f,g e A, i,X > 0 and a,p< 1. If f e S*(p) and g e K(a), then G(z) = f0zt • (g'{t)fdt is in the class JCp(j), where 7 = 1 - /x(l - /3) - A(1 - a).

If we consider i1 = |2 = ■ ■ ■ = ln = l and A1 = X2 = ■ ■ ■ = Xn = X in Theorem 2.1 results:

Corollary 2.3. Let fi,gi g Ap, n, X > 0 and «¿,/3» < 1, for i = l,n. If fi e S*(Pi) and 9i g ¡Cp(oi), then Gp,n(z) = f0 tp^1 ft • dt g Kp{7), where 7 =

1 — (1 — pi) — X£ (1 — ai).

i=1 i=1

Theorem 2.4. Let fi, gi g Ap, on, A > 1 and m, Aj > 0 for i = 1, n. If fi g Mp(Pi) and

nn

gi e Np(a.i), then Gp,n(z) e Np(y), where y =1 — J2 li(Pi — 1) — 2 Xi(ai — 1).

i=1 i=1

Proof. The proof is analogue with the proof of Theorem 2.1. □

Theorem 2.5. Let fi, gi g Ap, on, A < 1 and m, Aj > 0 for i = 1, n. If fi g S*(a, on) and

n n _

gi g K,p(a, Pi), then GPtTi g K,{a, 7), where 7 = 1 - j] mPi - j] ajoj, for i = l,n.

i=1 i=1

Proof. Using the idea from the proof of Theorem 2.1 and the definition of the class Kp (a,Y) we obtain that

nn

> 1 — liPi — Xiai.

i=1 i=1

□

For p = n = 1 in Theorem 2.5 we obtain: Corollary 2.6. Let f,g eA,a,P< 1 and i,X> 0. If f e S*(a,a) and g e K(a,p), then G(z) = f0z t (^ff ' (f'(t))A dt e /C(a, 7), where 7 = 1 - up - Xa.

Theorem 2.7. Let m, Xi positive real numbers and fi, gi g Ap for i = 1, n. If fi satisfies

zf'(z) 1

Re JtK ' < v h__

Mz) <P+ "

and gi satisfies

li

i=1

' zg''(z) \ 3

' 4 £ X,

i=1

then Gp,n is p-valently starlike in the open unit disk.

Proof. From the definition of Gp,n and from the hypotheses of our theorem it follows:

Gp,n(z ) \ i=1 i=1 / i = i fi(z )

i=i

n n

<p 1-J2 ^ -J2Xi Vi

i=i i=i

i=1

i=i

( \

i

p+——

. Evi/

\ i=1 /

^ Ai

g'i(z ) (

p--

i=i

\

4E Ai,

i=i /

<

<P + T

According to Lemma 1.1 we obtain that Gpn is in the class of p-valently starlike functions.

□

If in Theorem 2.7 we consider n = p =1, then we obtain: Corollary 2.8. Let X positive real numbers and f,g gA. If f satisfies

and g satisfies

Re

f (z )

zg"(z)

g'(z)

v

1 < i-

4A'

then G(z) = fgt (^t^) ' W{t))X dt is starlike in the open unit disk.

Theorem 2.9. Let i^i, Aj positive real numbers and fi,gi g Ap for i = 1, n. If the functions fi satisfies

zfi(z)

fi(z)

<

p

E Vi

i=i

and the functions gi satisfies

zg?(z)

g't(.z )

- p +1

<

E Ai

for z eU, then Gp,n is p-valently starlike in U.

Proof. Using the hypotheses of these theorem, we obtain that

zGP,n(z)

G'p,n (z)

+ 1 - p

= i n

<J2vi

i=i

<p+1.

fi(z)

zfl(z)

fi(z )

+ Ai

, g'i (z) zg'i(z)

<

gi (z)

- p +1

<

Using Lemma 1.2, results that Gpn is p-valently starlike in U. 396

□

3

3

p

1

For p = n = 1 in Theorem 2.9 results: Corollary 2.10. Let i,X positive real numbers and f,g eA. If the functions f satisfies

1

and the functions g satisfies

f(z)

zg''(z)

< l

g'(z)

1

< A

for zeU, then G(z) = J" t (^f-J • (g'{t)f dt is starlike in U.

Theorem 2.11. Let m, Aj positive real numbers and fi, gi g Ap for i = 1, n. If fi satisfies

fi(z)

(1 + a)(1 — b)J2 li

i=1

and gi satisfies

gi(z)

(1 + a)(1 — b)J2 Xi

i=1

for a > 0,b > 0,a + 2b < 1 and z eU, then Gp,n is p-valently close-to-convex in U. Proof. The proof is similar with the proof of Theorem 2.7, but we use Lemma 1.3. □

Considering p = n = 1 in Theorem 2.11 results: Corollary 2.12. Let i,X positive real numbers and f,g eA. If f satisfies

x*zM<i + - n

f (z) (1 + a)(1 — b)i

and g satisfies ( \

g'(z )J (1 + a)(1 — b)X

for a > 0, b > 0, a + 2b < 1 and z G U, then G(z) = J* t (^Y ■ (g'{t)f dt is close-to-convex in U.

Theorem 2.13. Let m, Aj positive real numbers and fi, gi g Ap for i = 1, n. If fi satisfies

r, zf'(z) 1

and gi satisfies

li

i=1

3 Xi

i=1

for z eU, then Gp,n is uniformly p-valently close-to-convex in U.

a

b

Proof. The proof is similar to Theorem 2.7, using Lemma 1.4.

For p = n = 1 in Theorem 2.13 we obtain: Corollary 2.14. Let i,X positive real numbers and f,g eA. If f satisfies

□

ReZ-m<l+1-

f (z)

and g satisfies

for zeM, then G(z) = f*t ' (9'(t))X dt is uniformly close-to-convex in U.

Bibliography

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10. Stanciu L., Ularu N. Some properties for two p-valent integral operators, to appear.

11. Uralegaddi A., Ganigi M.D., Sarangi S.M. Univalent functions with positive coefficients // Tamkang J. Math. Vol. 25(3). 1994. P. 225-230.

12. Ularu N., Breaz D., Frasin B. A. Two integral operators on the class N(fi) // Computers and Mathematics with Applications. Vol.62. 2011. P. 2551-2554.

Статья поступила в редакцию 27 марта 2014 г.

Сведения об авторах

Николета Улау — PhD; nicoletaularu@yahoo.com Лаура Станчи — PhD; laura_stanciu_30@yahoo.com

СВОЙСТВА ИНТЕГРАЛЬНОГО ОПЕРАТОРА p-ЗНАЧНЫХ ФУНКЦИЙ

Николета Улау, Лаура Станчи

Целью статьи является получение новых достаточных условий для оператора на классах звездообразных и выпуклых функций порядка a и типа а, класса p-значных звездообразные функций, p-значных выпуклых функций и однородно p-значных звездообразных и выпуклых функций. Биб-лиогр. 12 назв.

Ключевые слова: аналитические функции, функции близкие к выпуклым, функции близкие к звездообразным, интегральный оператор.