Argument estimates of certain multivalent analytic functions defined by integral operators *.

1. Introduction

Let A(p) denote the class of functions of the form:

f(z) = [z.sup.p] + [[infinity].summation over (k = [p+1])] [a.sub.k][z.sup.k], (p [member of] N), (1.1)

which are analytic in the open unit disc [DELTA] := {z: [absolute value of z] < 1}. For two functions f(z) and g(z) [member of] A(p), the Hadamard product (or convolution) is defined by

(f * g)(z):= [z.sup.p] + [[infinity].summation over (k = [p+1])] [a.sub.k][b.sub.k][z.sup.k] =: (g * f)(z), (1.2)

where

g(z) = [z.sup.p] + [[infinity].summation over (k = [p+1])][b.sub.k][z.sup.k](p [member of] N). (1.3)

For f(z) [member of] A(p), we consider following p-modification of the familiar Jung-Kim-Srivastava integral operator:

[I.sup.[sigma]] f(z) = [[[(p + 1)].sup.[sigma]]/[z[GAMMA]([sigma])]] [z.[integral].(0)] [(log [z/t]).sup.[[sigma]-1]] f(t)dt, (1.4)

= [z.sup.p] + [[infinity].summation over (n = [p+1])] [[([p+1]/[n+1])].sup.[sigma]] [a.sub.n][z.sup.n] [sigma] > 0. (1.5)

Obviously

[I.sup.0]f(z) [equivalent to] f(z). (1.6)

For the p-modified Jung-Kim-Srivastava integral operator, we easily get

z[[I.sup.[sigma]]f(z)]' = (p + 1)[I.sup.[sigma]-1]f(z) - [I.sup.[sigma]]f(z). (1.7)

Many classes of analytic functions defined by the p-modified Jung-Kim-Srivastava integral operator (1.4) were studied earlier by Shams et al. (6), Liu (4) and Patel and Mohanty (5).

In this paper, we derive certain argument properties of analytic functions defined by means of the p-modified Jung-Kim-Srivastava integral operator (1.4). In order to prove our main results, we shall require the following result.

Lemma 2.1 [3]. Let p(z) be analytic in [DELTA], with p(0) = 1, and p(z) [not equal to] 0 (z [member of] [DELTA]). Further suppose that [alpha], [beta] [member of] [R.sub.+] and

|arg(p(z) + [beta]zp'(z))| < [[pi]/2]([alpha] + [2/[pi]][tan.sup.-1][beta]), ([alpha] > 0,[beta] > 0), (2.1)

then

|arg(p(z))| < [[pi]/2][alpha] for z [member of] [DELTA]. (2.2)

2. Main Results

Theorem 3.1. If f(z) [member of] A(p) satisfies the condition

|[{[[[I.sup.[sigma]] f(z)]/[[I.sup.[sigma]] g(z)]]}.sup.[gamma]]{1 + [[lambda]/p] ([[I.sup.[sigma]-1]f(z)]/[[I.sup.[sigma]]f(z)] - [[I.sup.[sigma]-1]g(z)]/[[I.sup.[sigma]]g(z)])}| < [[pi]/2][alpha] + [tan.sup.-1]([lambda]/[p(p + 1)][alpha]), (3.1)

then

|[{[[[I.sup.[sigma]]f(z)]/[[I.sup.[sigma]]g(z)]]}.sup.[gamma]]| < [[pi]/2][alpha] (3.2)

where [alpha], [beta], [gamma], [sigma] [member of] [R.sub.+], [lambda] [greater than or equal to] 0 and z [member of] [DELTA].

Proof. Define a function

p(z) = [{[[[I.sup.[sigma]]f(z)]/[[I.sup.[sigma]]g(z)]]}.sup.[gamma]], [gamma][not equal to]0 (3.3)

then p(z) = 1 + [c.sub.1]z + [c.sub.2][z.sup.2] + ........ which is analytic in [DELTA] with p(0) = 1 and p(z) [not equal to] 0 (z [member of] [DELTA]).

Diffrentiating (2.3) logarithmically, we get

[[zp'(z)]/[p(z)]] = [gamma][[z[[I.sup.[sigma]]f(z)]']/[[I.sup.[sigma]]f(z)]-[z[[I.sup.[sigma]]g(z)]']/[[I.sup.[sigma]]g(z)]]. (3.4)

Now making use of identity (1.7) in (3.4), we easily get

p(z) + [[lambda]/[[gamma]p(p+1)]]zp'(z) = [{[[[I.sup.[sigma]]f(z)]/[[I.sup.[sigma]]g(z)]]}.sup.[gamma]]{1 + [[lambda]/p]([[I.sup.[sigma]-1]g(z)]/[[I.sup.[sigma]]g(z)]-[[I.sup.[sigma]-1]f(z)]/[[I.sup.[sigma]]f(z)])}, (3.5)

and the statement of the Theorem 3.1 directly follows from Lemma 2.1.

Setting [gamma] = 1 and g(z) = [z.sup.p] i.e. all [b.sub.i] = 0 (i = p + 1, ......) in Theorem 3.1, we easily arrive at the

Corollary 3.2. If f(z) [member of] A(p) satisfies

|arg {[[lambda]/p] [[[I.sup.[[sigma]-1]]f(z)]/[z.sup.p]] + [[(p-[lambda])]/p] [[[I.sup.[sigma]] f(z)]/[z.sup.p]]}| < [[pi]/2][alpha] + [tan.sup.-1]([lambda]/[p(p + 1)][alpha]), (3.6)

then

|arg ([[I.sup.[sigma]] f(z)]/[z.sup.p])| < [[pi]/2][alpha], (3.7)

where [alpha], [beta], [sigma] [member of] [R.sub.+], [lambda] [greater than or equal to] 0 and z [member of] [DELTA].

Again taking [gamma] = p = 1, we get

Corollary 3.3. Let [alpha], [sigma] [member of] [R.sub.+] and [lambda] [greater than or equal to] 0. If f(z) [member of] A(1) satisfies

|arg {[lambda][[[I.sup.[[sigma]-1]] f(z)]/z] + (1-[lambda]) [[[I.sup.[sigma]]f(z)]/z]}| < [[pi]/2][alpha] + [tan.sup.-1]([lambda]/2[alpha]), (3.8)

then

|arg ([[I.sup.[sigma]]f(z)]/z)| < [[pi]/2][alpha]. (3.9)

Further taking [gamma] = 1, [lambda] = p + 1 and [sigma] [right arrow] 0 in Theorem 3.1, we get result on argument estimate given earlier by Cho et al. (1).

If we put [lambda] = p = 1, and let [sigma] [right arrow] 0 in Theorem 3.1, and replace [lambda] by 2[beta] therein, we get a result due to Lashin (3).

Lastly taking [gamma] = 1, and f(z) = [z.sup.p] i.e. ([a.sub.i] = 0, i = p+1, ......) in Theorem 3.1, we get an interesting result contained in

Corollary 3.4. Let [[z.sup.p]/[I.sup.[sigma]]g(z)]] [not equal to] 0, g(z) [member of] A(p) and [lambda] [greater than or equal to] 0. Suppose that

|arg [(1 + [lambda]/p)[[z.sup.p]/[[I.sup.[sigma]]g(z)]]-[[lambda]/p][[[I.sup.[sigma]-1]g(z)]/[[I.sup.[sigma]]g(z)]]([z.sup.p]/[[I.sup.[sigma]]g(z)])]|<[[pi]/2][alpha] + [tan.sup.-1]([lambda]/[p(p + 1)][alpha]), (3.10)

then

|[[z.sup.p]/[[I.sup.[sigma]]g(z)]]| < [[pi]/2][alpha]. ([alpha][member of][R.sub.+], z[member of] [DELTA]) (3.11)

Theorem 3.5. Let [lambda], [sigma] [member of] [R.sub.+] and 0 < [lambda] < p. Suppose that f(z) [member of] A(p) satisfies

|[[[I.sup.[sigma]]f(z)]/[z.sup.p]]|<[[pi]/2][alpha] + [tan.sup.-1]([[lambda][alpha]]/[p(p + 1)]) (3.12)

then we have

|arg([p(p + 1)]/[lambda][Z.sup.[-p(p + 1)/[lambda]]] [z.[integral].0][ [t.sup.[(p+1)(p-[lambda])/[lambda]]][I.sup.[sigma]] f(t)dt)| < [[pi]/2][alpha]. (3.13)

Proof. Consider the function

p(z) = ([p(p + 1)]/[lambda][Z.sup.[-p(p + 1)/[lambda]]][z.[integral].0][t.sup.[(p+z)(p-[lambda])/[lambda]]][I.sup.[sigma]]f(t)dt). (3.14)

Obviously

p(z) = 1 + [c.sub.1]z + [c.sub.2][z.sup.2] + ............ (3.15)

and p(z) is an analytic in [DELTA]. Also p(0) = 1, and p'(z) [not equal to] 0.

Differentiating (3.14), we get the following result after some computations

p(z) + [[lambda]/[p(p + 1)]]zp'(z) = [[[I.sup.[sigma]]f(z)]/[z.sup.p]]. (3.16)

Now making use of Lemma 2.1, the proof of the Theorem 3.5 is complete.

Setting p = 1, [lambda] = 2 and [sigma] [right arrow] 0, in Theorem 3.5, we arrive at the following interesting result contained in

Corollary 3.6. Let f(z) [member of] A(1) satisfies

|arg([f(z)]/z)| < [[pi]/2][alpha] + [tan.sup.-1]([alpha]), (3.17)

then we have

|arg([1/z] [z.[integral].0] [[f(t)]/t] dt)| < [[pi]/2][alpha]([alpha]>0 and z [member of] [DELTA]). (3.18)

Acknowledgements

The authors are thankful to Professor H. M. Srivastava (University of Victoria, Canada) for his kind help and constant encourgement. The second author is thankful to the Council of Scientific and Industrial Research of the Government of India for providing Junior Research Fellowship under Research Scheme No. 09/135(0434)/2006-EMR-1.

References

(1) N. E. Cho, J. A. Kim, I. H. Kim, and S. H. Lee, Angular estimates of certain multivalent functions, Math. Japon., 50(1999), 359-370.

(2) N. E. Cho, J. Patel, and G. P. Mohapatra, Angular estimates of certain multivalent functions involving a linear operator, Internat. J. Math. Math. Sci., 31(2002), 659-673.

(3) A. Y. Lashin, Applications of Nunokawa's theorem J. Inequal. Pure Appl. Math., 5(4)(2004), Article 111 (electronic).

(4) J.-L. Liu, Notes on Jung-Kim-Srivastava integral operator, J. Math. Anal. Appl., 294(2004), 96-103.

(5) J. Patel and A. K. Mohanty, On a class of p-valent analytic functions with complex order, Kyungpook Math. J., 43(2003), 199-209.

(6) S. Shams, S. R. Kulkarni, and J. M. Jahangiri, Subordination properties of p-valent functions defined by integral operators, Internat. J. Math. Math. Sci., 2006(2006), 1-3, Article ID 94572.

S. P. Goyal [dagger] and Pranay Goswami [double dagger]

Department of Mathematics, University of Rajasthan Jaipur 302004, Rajasthan, India

Received November 27, 2007, Accepted January 11, 2008.

* 2000 Mathematics Subject Classification. Primary 30C45.

[dagger] E-mail: [e-mail]

[double dagger] E-mail: [e-mail]