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In this article we provide a vertical heightestimate for compact special Weingarten surfaces of elliptictype in ²()×ℝ, i.e. surfaces whose mean curvature ℍ andextrinsic Gauss curvature ₑ satisfy ℍ=(ℍ²-ₑ) with4(’())²<1, for all ∈[0,+∞). The vertical height estimategeneralizes a result by Rosenberg and Sa Earp and applies onlyto surfaces verifying a height estimate condition. When <0,using also a horizontal height estimate, we show a non-existenceresult for properly embedded Weingarten surfaces of elliptictype in ℍ²()×ℝ with finite topology and one end

Height estimate for special Weingarten surfaces of elliptic type in 𝕄²(𝕔)×ℝ