Communications in Mathematical Analysis Volume 6, Number 2, pp. 21–32 (2009) ISSN 1938-9787 www.commun-math-anal.org O N C ERTAIN R ESULTS OF D IFFERENTIAL S UBORDINATION AND S UPERORDINATION S UKHWINDER S INGH ∗ Department of Applied Sciences B.B.S.B.Engineering College Fatehgarh Sahib-140 407, Punjab, India S USHMA GUPTA † Department of Mathematics S.L.I.E.T. Longowal-148 106, Punjab, India S UKHJIT S INGH ‡ Department of Mathematics S.L.I.E.T. Longowal-148 106, Punjab, India (Communicated by Toka Diagana) Abstract Let α, β, γ and δ be complex numbers such that β, δ = 0. Define Φ on D = C −{0, − γ β } as Φ(w , zw ′ ; z)=(w) α  w + zw ′ βw + γ  δ , z ∈ E, where E = {z : |z| < 1}. We find the sufficient conditions for analytic function p with p(E) ⊂ D and analytic univalent functions q 1 and q 2 , with q 1 (E) and q 2 (E) contained in D, such that Φ(q 1 (z), zq ′ 1 (z); z) ≺ Φ( p(z), zp ′ (z); z) ≺ Φ(q 2 (z), zq ′ 2 (z); z), implies q 1 (z) ≺ p(z) ≺ q 2 (z), and that q 1 and q 2 are, respectively, best subordinant and best dominant. We give applications of these results to univalent, φ-like and P -valent functions and show that these results unify a number of previously known results. ∗ E-mail address: ss billing@yahoo.co.in † E-mail address: sushmagupta1@yahoo.com ‡ E-mail address: sukhjit d@yahoo.com