GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 5, 995, 535-545 PROPERTIES OF CERTAIN INTEGRAL OPERATORS SHIGEYOSHI OWA Abstract. Two integral operators P α and Q α for analytic functions in the open unit disk are introduced. The object of the present paper is to derive some properties of integral operators P α and Q α.. Introduction Let A be the class of functions of the form = z + a n z n (.) which are analytic in the open unit disk U = z : z <. cently, Jung, Kim, and Srivastava [] have introduced the following one-parameter families of integral operators: and P α f = P α = Q α f = Q α = J α f = J α = α + z α n=2 z ( 2α log z ) α f(t)dt (α > 0), (.2) zγ(α) t 0 ( α + ) z α ( z t ) α t f(t)dt (.3) z 0 (α > 0, > ) z 0 t α f(t)dt (α > ), (.4) 99 Mathematics Subject Classification. 30C45. Key words and phrases. Analytic functions, logarithmic differentiation, integral operator. 535 072-947X/95/900-0535$7.50/0 c 995 Plenum Publishing Corporation
536 SHIGEYOSHI OWA where Γ(α) is the familiar Gamma function, and (in general) [ ] [ ] α Γ(α + ) α = Γ(α + )Γ( + ) =. (.5) α For α N =, 2, 3,..., the operators P α, Q α, and J α were considered by Bernardi ([2], [3]). Further, for a real number α >, the operator J α was used by Owa and Srivastava [4], and by Srivastava and Owa ([8], [6]). mark. For A given by (.), Jung, Kim, and Srivastava [] have shown that ( 2 ) αan P α = z + z n (α > 0), (.6) n + n=2 Q α Γ(α + + ) Γ( + n) = z + Γ( + ) Γ(α + + n) a nz n (.7) and J α = z + n=2 (α > 0, > ) n=2 By virtue of (.6) and (.8), we see that ( α + ) a n z n (α > ). (.8) α + n J α = Q α (α > ). (.9) 2. An Application of the Miller Mocanu Lemma To derive some properties of operators, we have to recall here the following lemma due to Miller and Mocanu [7]. Lemma. Let w(u, v) be a complex valued function, w : D C, D C C (C is the complex plane), and let u = u + iu 2 and v = v + iv 2. Suppose that the function w(u, v) satisfies the following conditions: (i) w(u, v) is continuous in D; (ii) (, 0) D and w(, 0) > 0; (iii) w(iu 2, v ) 0 for all (iu 2, v ) D and such that v ( + u 2 2)/2. Let p(z) be regular in U and p(z) = + p z + p 2 z 2 +... such that (p(z), zp (z)) D for all z U. If w(p(z), zp (z)) > 0 (z U), then p(z) > 0 (z U). Applying the above lemma, we derive
PROPERTIES OF CERTAIN INTEGRAL OPERATORS 537 Theorem. If A satisfies for some ( < ), then Proof. Noting that we have P P α > (α > 2; z U) (2.) P α P α > 4 + 6 2 8 + 7 8 (z U). (2.2) z(p α ) = 2P α F (z) P α (α > ), (2.3) z(p α ) P α Define the function p(z) by with P α P α = 2 P α P α (α > ). (2.4) = γ + ( γ)p(z) (2.5) γ = 4 + 6 2 8 + 7. (2.6) 8 Then p(z) = + p z + p 2 z 2 +... is analytic in the open unit disk U. Since or we have z(p α ) P α z(p α ) P α = ( γ)zp (z) γ + ( γ)p(z), (2.7) P P α = P α ( γ)zp (z) P α + 2γ + ( γ)p(z), (2.8) = γ + ( γ)p(z) + P P α = Therefore, if we define the function w(u, v) by then we see that w(u, v) = γ + ( γ)u + ( γ)zp (z) 2γ + ( γ)p(z) > 0. (2.9) ( γ)v 2γ + ( γ)u, (2.0)
538 SHIGEYOSHI OWA ( (i) w(u, v) is continuous in D = C γ γ ) C; (ii) (, 0) D and w(, 0) = > 0; (iii) for all (iu 2, v ) D and such that v ( + u 2 2)/2, w(iu 2, v ) = γ + γ( γ)v 2γ 2 + ( γ) 2 u 2 2 γ γ( γ)( + u2 2) 4γ 2 + ( γ) 2 u 2 2 = ( γ)(γ + 4( γ)) = 4γ 2 + ( γ) 2 u2 2 u2 2 0. This implies that the function w(u, v) satisfies the conditions in Lemma. Thus, applying Lemma, we conclude that P α P α > γ = = 4 + 6 2 8 + 7 8 Taking the special values for in Theorem, we have Corollary. Let be in the class A. Then P (i) P α > (z U) 2 P α = P α > (z U), 4 P (ii) P α > (z U) 4 P α 5 = P α > (z U), 4 P (iii) P α > 0 (z U) P α 7 = P α > (z U), 8 P (iv) P α > (z U) 4 P α = P α > (z U), 2 and P (v) P α > (z U) 2 P α 7 + = P α > (z U). 8 (z U). (2.)
Next, we have PROPERTIES OF CERTAIN INTEGRAL OPERATORS 539 Theorem 2. If A satisfies > γ (α > 2, > ; z U) (2.2) for some γ ((α + 3)/2(α + ) γ < ), then where α Q α > δ (z U), (2.3) δ = + 2γ(α + ) + ( + 2γ(α + )) 2 + 8(α + ). (2.4) 4(α + ) Proof. By the definition of Q α, we know that z(q α ) = (α + ) (α + )Q α (2.5) (α >, > ), so that z(q α ) Q α = (α + ) Qα Q α (α + ). (2.6) We define the function p(z) by Q α = δ + ( δ)p(z). (2.7) Then p(z) = + p z + p 2 z 2 +... is analytic in U. Making use of the logarithmic differentiations in both sides of (2.7), we have z( ) Applying (2.5) to (2.8), we obtain that Q = + ( δ)zp (z) δ + ( δ)p(z) = z(qα ) Q α + ( δ)zp (z) δ + ( δ)p(z). (2.8) α + = +(α + )( δ)p(z) + (α + ) Qα Q α + α + δ(α + ) + ( δ)zp (z) δ + ( δ)p(z), (2.9)
540 SHIGEYOSHI OWA that is, that Now, we let γ = +(α + )( δ)p(z) + w(u, v) = +(α + )( δ)u + α + δ(α + ) + ( δ)zp (z) γ > 0. (2.20) δ + ( δ)p(z) δ(α + ) + α + ( δ)v δ + ( δ)u γ (2.2) with u = u + iu 2 and v = v + iv 2 (. Then w(u, v) satisfies that (i) w(u, v) is continuous in D = C δ δ ) C; (ii) (, 0) D and w(, 0) = γ > 0; (iii) for all (iu 2, v ) D and such that v ( + u 2 2)/2, w(iu 2, v ) = δ(α + ) + α + + δ( δ)v δ 2 + ( δ) 2 u 2 γ δ(α + ) 2 α + γ(α + ) δ( δ)( + u2 2) 2δ 2 + ( δ) 2 u 2 2 0. Thus, the function w(u, v) satisfies the conditions in Lemma. This shows that p(z) > 0, or then α Q α > δ (z U). (2.22) If we take γ = (α + 3)/2(α + ) in Theorem 2, then we have Corollary 2. If A satisfies > α + 3 2(α + ) (α > 2, > ; z U), α Q α > 2 (z U). Further, letting α = 2 and γ = /2 in Theorem 2, we have
then PROPERTIES OF CERTAIN INTEGRAL OPERATORS 54 Corollary 3. If A satisfies Q Q 2 > 2 ( > ; z U), > + 5 4 (z U). 3. An Application of Jack s Lemma We need the following lemma due to Jack [8], (also, due to Miller and Mocanu [7]). Lemma 2. Let w(z) be analytic in U with w(0) = 0. If w(z) attains its maximum value on the circle z = r < at a point z 0 U, then we can write where k is a real number and k. then z 0 w (z 0 ) = kw(z 0 ), (3.) Applying Lemma 2 for the operator P α, we have Theorem 3. If A satisfies where P P α > (α > 2, /4; z U), (3.2) Proof. Defining the function w(u, v) by P α P α > γ (z U), (3.3) γ = 3 + 4 + 6 2 40 + 9. (3.4) 8 P α P α = ( 2γ)w(z), (3.5) + w(z) we see that w(z) is analytic in U and w(0) = 0. It follows from (3.5) that z(p α ) P α = z(p α ) P α ( 2γ)zw (z) ( 2γ)w(z) zw (z) + w(z). (3.6)
542 SHIGEYOSHI OWA Using (2.3), we obtain that 2 P ( 2γ)w(z) P α = + w(z) zw (z) ( ( 2γ)w(z) w(z) ( 2γ)w(z) + w(z) ). (3.7) + w(z) If we suppose that there exists a point z 0 U such that then Lemma 2 gives us that max w(z) = w(z 0) = (w(z 0 ) ), z z 0 z 0 w (z 0 ) = kw(z 0 ) (k ). Therefore, letting w(z 0 ) = e iθ (θ π), we have P f(z 0 ) ( 2γ)w(z0 ) P α = f(z 0 ) + w(z 0 ) k ( ( 2γ)w(z0 ) 2 ( 2γ)w(z 0 ) + w(z ) 0) = + w(z 0 ) ( 2γ)e iθ = + e iθ k ( ( 2γ)e iθ ) eiθ + 2 ( 2γ)eiθ + e iθ = = γ k ( ( 2γ)(cos θ ( 2γ)) 2 + ( 2γ) 2 2( 2γ) cos θ + ) 2 γ k ( 2 2 2γ ) γ(3 4γ) =. (3.8) 2( γ) 4( γ) This contradicts our condition (3.2). Therefore, we have P α w(z) = P α P α < (z U), (3.9) P α + ( 2γ) which implies (3.3). Taking = 0 and = /4 in Theorem 3, we have Corollary 4. Let be in the class A. Then P P α > 0 (α > 2; z U) P α = P α > 3 4 (z U)
and PROPERTIES OF CERTAIN INTEGRAL OPERATORS 543 Finally, we prove P P α > (α > 2; z U) 4 P α = P α > 2 Theorem 4. If A satisfies for some γ (γ < ), then (z U). > γ (α > 2; > ; z U) (3.0) α Q α > δ (z U), (3.) where δ (0 δ < ) is the smallest positive root of the equation 2(α + )δ 2 2(α + )(γ + ) (2γ )δ + 2(α + )γ + = 0. (3.2) Proof. Defining the function w(z) by we obtain that Q α = Q = zw (z) w(z) ( 2δ)w(z), (3.3) + w(z) ( 2δ)w(z) (α + ) + w(z) α + ( ( 2δ)w(z) ( 2δ)w(z) + w(z) + w(z) Therefore, supposing that there exists a point z 0 U such that we see that max w(z) = w(z 0) = (w(z 0 ) ), z z 0 f(z 0 ) f(z 0 ) (α + )δ α + Making γ = 0 in Theorem 4, we have δ 2( δ) ). (3.4) = γ. (3.5)
544 SHIGEYOSHI OWA Corollary 5. If A satisfies then > 0 (α > 2; > ; z U), (3.6) α Q α > α + (z U). (3.7) Letting γ = /2 in Theorem 4, we have Corollary 6. If A satisfies then > 2 α Q α > α + 3 α + (α > 2; > ; z U), (3.8) (z U). (3.9) Further, if α = 3, we have Q 2 > 2 = ( > ; z U) 2 Q 3 Acknowledgement > 0 (z U). This research was supported in part by the Japanese Ministry of Education, Science and Culture under a Grant-in-Aid for General Scientific search (No. 04640204). ferences. S. D. Bernardi, Convex and starlike univalent functions. Trans. Amer. Math. Soc. 35 (969), 429-446. 2. S. D. Bernardi, The radius of univalence of certain analytic functions. Proc. Amer. Math. Soc. 24 (970), 32-38. 3. I. S. Jack, Functions starlike and convex of order α, J. London Math. Soc. 3 (97), 469-474.
PROPERTIES OF CERTAIN INTEGRAL OPERATORS 545 4. I. B. Jung, Y. C. Kim, and H. M. Srivastava, The Hardy space of analytic functions associated with certain one-parameter families of integral operators, to appear. 5. S. S. Miller and P. T. Mocanu, Second-order differential inequalities in the complex plane. J. Math. Anal. Appl. 65 (978), 289-305. 6. S. Owa and H. M. Srivastava, Some applications of the generalized Libera operator. Proc. Japan Acad. 62 (986), 25-28. 7. H. M. Srivastava and S. Owa, New characterization of certain starlike and convex generalized hypergeometric functions. J. Natl. Acad. Math. India 3 (985), 98-202. 8. H. M. Srivastava and S. Owa, A certain one-parameter additive family of operators defined on analytic functions. J. Anal. Appl. 8 (986), 80-87. Author s address: Department of Mathematics Kinki University Higashi-Osaka, Osaka 577 Japan (ceived 3.06.94)